Page 1 of 2
PROPERTIES
OF TRIANGLES
How can a goalkeeper best
defend the goal?
260
Page 2 of 2
CHAPTER
5
APPLICATION: Goalkeeping
Soccer goalkeepers
use triangle
relationships to help block goal
attempts.
An opponent can shoot the ball
from many different angles. The
goalkeeper determines the best
defensive position by imagining a
triangle formed by the goal posts
and the opponent.
A
B
C
X
Think & Discuss
Use the diagram for Exercises 1 and 2.
1. The opponent at X is trying to score a goal.
Which position do you think is best for the
goalkeeper, A, B, or C? Why?
2. Estimate the measure of ™X, known as the
shooting angle. How could the opponent change
positions to increase the shooting angle?
Learn More About It
INT
You will learn more about strategies of goalkeeping
in Exercises 33–35 on p. 270.
NE
ER T
APPLICATION LINK Visit www.mcdougallittell.com
for more information about angles and goalkeeping.
261
Page 1 of 1
Study Guide
CHAPTER
5
PREVIEW
What’s the chapter about?
Chapter 5 is about properties of triangles. In Chapter 5, you’ll learn how to
•
•
use properties of special lines and segments related to triangles.
compare side lengths and angle measures in one or more triangles.
K E Y VO C A B U L A RY
Review
• perpendicular bisector of a
• intersect, p. 12
• midpoint, p. 34
• angle bisector, p. 36
• perpendicular lines, p. 79
triangle, p. 272
• concurrent lines, p. 272
• circumcenter of a triangle,
p. 273
• angle bisector of a triangle,
p. 274
• incenter of a triangle, p. 274
New
• perpendicular bisector,
p. 264
PREPARE
• median of a triangle, p. 279
• centroid of a triangle, p. 279
• altitude of a triangle, p. 281
• orthocenter of a triangle,
p. 281
• midsegment of a triangle,
p. 287
• indirect proof, p. 302
Are you ready for the chapter?
SKILL REVIEW Do these exercises to review key skills that you’ll apply in this
chapter. See the given reference page if there is something you don’t understand.
Æ
Æ
1. Draw a segment and label it AB. Construct a bisector of AB. Label its
midpoint M. (Review p. 34)
2. Draw an angle and label it ™P. Construct an angle bisector of ™P. (Review p. 36)
Use the diagram at the right.
y
B(0, 4)
Æ
3. Find the coordinates of the midpoint of BC.
(Review p. 35)
1
Æ
4. Find the length of AB. (Review p. 19)
C(2, 0)
Æ
5. Find the slope of BC. (Review p. 165)
1
A(3, 0) x
Æ
6. Find the slope of a line perpendicular to BC.
(Review p. 174)
STUDY
STRATEGY
Here’s a study
strategy!
Check Your Memory
Without looking at your book
or you
list of important vocabulary ter r notes, write a
ms and skills. Then
look through the chapter and
your notes as you
compare them with your list.
Did you miss anything?
262
Chapter 5
Page 1 of 8
5.1
Perpendiculars and Bisectors
What you should learn
GOAL 1 Use properties of
perpendicular bisectors.
Use properties of
angle bisectors to identify
equal distances, such as the
lengths of beams in a roof
truss in Example 3.
GOAL 2
Why you should learn it
RE
FE
䉲 To solve real-life
problems, such as deciding
where a hockey goalie
should be positioned
in Exs. 33–35.
AL LI
GOAL 1
USING PROPERTIES OF PERPENDICULAR BISECTORS
In Lesson 1.5, you learned that a segment bisector
intersects a segment at its midpoint. A segment, ray, line,
or plane that is perpendicular to a segment at its midpoint
is called a perpendicular bisector.
C
given
segment
P
A
B
The construction below shows how to draw a line
perpendicular
that is perpendicular to a given line or segment at a
bisector
point P. You can use this method to construct a
perpendicular bisector of a segment, as described below
¯
˘
Æ
the activity.
CP is a fi bisector of AB .
ACTIVITY
Construction
Perpendicular Through a Point on a Line
Use these steps to construct a line that is perpendicular to a given line m and
that passes through a given point P on m.
C
C
m
m
A
1
P
B
Place the compass
point at P. Draw an
arc that intersects
line m twice. Label
the intersections as
A and B.
A
2
P
m
B
Use a compass setting
greater than AP. Draw
an arc from A. With
the same setting, draw
an arc from B. Label
the intersection of the
arcs as C.
A
3
P
Use a straightedge
¯
˘
to draw CP . This
line is perpendicular
to line m and passes
through P.
ACTIVITY CONSTRUCTION
STUDENT HELP
Look Back
For a construction of a
perpendicular to a line
through a point not on the
given line, see p. 130.
264
You can measure ™CPA on your construction to verify that the constructed
¯
˘ Æ
line is perpendicular to the given line m. In the construction, CP fi AB
¯
˘
Æ
and PA = PB, so CP is the perpendicular bisector of AB.
A point is equidistant from two points if its distance from each point is the
same. In the construction above, C is equidistant from A and B because C was
drawn so that CA = CB.
Chapter 5 Properties of Triangles
B
Page 2 of 8
¯
˘
Theorem 5.1 below states that any point on the perpendicular bisector CP in the
construction is equidistant from A and B, the endpoints of the segment. The
converse helps you prove that a given point lies on a perpendicular bisector.
THEOREMS
THEOREM 5.1
Perpendicular Bisector Theorem
C
If a point is on the perpendicular bisector of a
segment, then it is equidistant from the endpoints
of the segment.
¯
˘
Æ
If CP is the perpendicular bisector of AB,
then CA = CB.
THEOREM 5.2
P
A
CA = CB
Converse of the Perpendicular
Bisector Theorem
If a point is equidistant from the endpoints of a
segment, then it is on the perpendicular bisector
of the segment.
C
P
A
B
D
If DA = DB, then D lies on the
Æ
bisector of AB .
T H E Operpendicular
REM
Proof
B
¯
˘
D is on CP.
Plan for Proof of Theorem 5.1 Refer to the diagram for Theorem 5.1 above.
¯
˘
Æ
Suppose that you are given that CP is the perpendicular bisector of AB. Show that
right triangles ¤APC and ¤BPC are congruent using the SAS Congruence
Æ
Æ
Postulate. Then show that CA £ CB.
Exercise 28 asks you to write a two-column proof of Theorem 5.1 using
this plan for proof. Exercise 29 asks you to write a proof of Theorem 5.2.
EXAMPLE 1
Logical
Reasoning
Using Perpendicular Bisectors
¯
˘
Æ
In the diagram shown, MN is the perpendicular bisector of ST .
T
a. What segment lengths in the diagram are equal?
12
¯
˘
b. Explain why Q is on MN .
M
¯
˘
q
N
SOLUTION
12
Æ
a. MN bisects ST , so NS = NT. Because M is on the
S
Æ
perpendicular bisector of ST , MS = MT (by Theorem 5.1).
The diagram shows that QS = QT = 12.
b. QS = QT, so Q is equidistant from S and T. By Theorem 5.2, Q is on the
Æ
¯
˘
perpendicular bisector of ST , which is MN .
5.1 Perpendiculars and Bisectors
265
Page 3 of 8
GOAL 2
USING PROPERTIES OF ANGLE BISECTORS
The distance from a point to a line is defined as the length
of the perpendicular segment from the point to the line. For
instance, in the diagram shown, the distance between the
point Q and the line m is QP.
q
m
P
When a point is the same distance from one line as it is from
another line, then the point is equidistant from the two lines
(or rays or segments). The theorems below show that a point in
the interior of an angle is equidistant from the sides of the angle
if and only if the point is on the bisector of the angle.
THEOREMS
THEOREM 5.3
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is
equidistant from the two sides of the angle.
B
D
A
If m™BAD = m™CAD, then DB = DC.
C
DB = DC
THEOREM 5.4
Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is
equidistant from the sides of the angle, then it
lies on the bisector of the angle.
If DB = DC, then m™BAD = m™CAD.
B
D
A
C
m™BAD = m™CAD
THEOREM
A paragraph proof of Theorem 5.3 is given in Example 2. Exercise 32 asks you
to write a proof of Theorem 5.4.
EXAMPLE 2
Proof
Proof of Theorem 5.3
GIVEN 䉴 D is on the bisector of ™BAC.
Æ
Æ˘ Æ
Æ˘
B
DB fi AB , DC fi AC
PROVE 䉴 DB = DC
D
A
Plan for Proof Prove that ¤ADB £ ¤ADC.
Æ
Æ
Then conclude that DB £ DC, so DB = DC.
C
SOLUTION
Paragraph Proof By the definition of an angle bisector, ™BAD £ ™CAD.
Because ™ABD and ™ACD are right angles, ™ABD £ ™ACD. By the Reflexive
Æ
Æ
Property of Congruence, AD £ AD. Then ¤ADB £ ¤ADC by the AAS
Congruence Theorem. Because corresponding parts of congruent triangles are
Æ
Æ
congruent, DB £ DC. By the definition of congruent segments, DB = DC.
266
Chapter 5 Properties of Triangles
Page 4 of 8
FOCUS ON
EXAMPLE 3
CAREERS
Using Angle Bisectors
ROOF TRUSSES Some roofs are built
with wooden trusses that are assembled
in a factory and shipped to the building
site. In the diagram of the roof truss
Æ˘
shown below, you are given that AB
bisects ™CAD and that ™ACB and
™ADB are right angles. What can you
Æ
Æ
say about BC and BD?
RE
FE
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AL I
ENGINEERING
TECHNICIAN
A
C
INT
In manufacturing,
engineering technicians
prepare specifications for
products such as roof
trusses, and devise and run
tests for quality control.
D
B
NE
ER T
CAREER LINK
www.mcdougallittell.com
SOLUTION
Æ
Æ
Æ
Æ
Because BC and BD meet AC and AD at right angles, they are perpendicular
segments to the sides of ™CAD. This implies that their lengths represent the
Æ˘
Æ
˘
distances from the point B to AC and AD. Because point B is on the bisector of
™CAD, it is equidistant from the sides of the angle.
䉴
Æ
Æ
So, BC = BD, and you can conclude that BC £ BD.
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Skill Check
✓
Æ
? of AB, then D is equidistant from A and B.
1. If D is on the 2. Point G is in the interior of ™HJK and is equidistant from the sides of the
Æ˘
Æ˘
angle, JH and JK . What can you conclude about G? Use a sketch to support
your answer.
¯
˘
Æ
In the diagram, CD is the perpendicular bisector of AB .
Æ
C
Æ
3. What is the relationship between AD and BD?
4. What is the relationship between ™ADC and
™BDC?
Æ
A
D
B
Æ
5. What is the relationship between AC and BC ?
Explain your answer.
Æ˘
In the diagram, PM is the bisector of ™LPN.
M
6. What is the relationship between ™LPM and
™NPM?
N
L
Æ˘
7. How is the distance between point M and PL
Æ˘
P
related to the distance between point M and PN ?
5.1 Perpendiculars and Bisectors
267
Page 5 of 8
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 811.
LOGICAL REASONING Tell whether the information in the diagram
Æ
allows you to conclude that C is on the perpendicular bisector of AB .
Explain your reasoning.
8.
9. A
C
8
A
P
C
10.
B
7
P
C
B
A
P
B
LOGICAL REASONING In Exercises 11–13, tell whether the information
in the diagram allows you to conclude that P is on the bisector of ™A.
Explain.
11.
12.
13.
7
4
P
3
A
8
P
P
7
A
8
A
Æ
CONSTRUCTION Draw AB with a length of 8 centimeters. Construct a
14.
perpendicular bisector and draw a point D on the bisector so that the distance
Æ
Æ
Æ
between D and AB is 3 centimeters. Measure AD and BD.
15.
CONSTRUCTION Draw a large ™A with a measure of 60°. Construct the
angle bisector and draw a point D on the bisector so that AD = 3 inches.
Draw perpendicular segments from D to the sides of ™A. Measure these
segments to find the distance between D and the sides of ™A.
USING PERPENDICULAR BISECTORS Use the diagram shown.
¯
˘
Æ
Æ
Æ
16. In the diagram, SV fi RT and VR £ VT . Find VT.
¯
˘
Æ
Æ
14
R
U
Æ
17. In the diagram, SV fi RT and VR £ VT . Find SR.
8
¯
˘
V
14
18. In the diagram, SV is the perpendicular bisector
Æ
of RT. Because UR = UT = 14, what can you
conclude about point U?
S
17
T
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 8–10, 14,
16–18, 21–26
Example 2: Exs. 11–13,
15, 19, 20, 21–26
Example 3: Exs. 31,
33–35
268
USING ANGLE BISECTORS Use the diagram shown.
Æ˘
Æ
19. In the diagram, JN bisects ™HJK, NP fi JP ,
Æ
Æ
Æ˘
20. In the diagram, JN bisects ™HJK, MH fi JH ,
Æ
Æ˘
MK fi JK , and MH = MK = 6. What can
you conclude about point M?
Chapter 5 Properties of Triangles
6
P
Æ˘
NQ fi JQ , and NP = 2. Find NQ.
Æ˘
H
Æ˘
2
N
J
M
q
6
K
Page 6 of 8
USING BISECTOR THEOREMS In Exercises 21–26, match the angle measure
or segment length described with its correct value.
A. 60°
B. 8
C. 40°
D. 4
E. 50°
F. 3.36
W
4
U
X
21. SW
22. m™XTV
23. m™VWX
24. VU
25. WX
26. m™WVX
27.
50ⴗ
3.36
30ⴗ
S
V
T
¯
˘
Æ
PROVING A CONSTRUCTION Write a proof to verify that CP fi AB in
the construction on page 264.
STUDENT HELP
28.
Look Back
For help with proving that
constructions are valid,
see p. 231.
PROVING THEOREM 5.1 Write a proof of Theorem 5.1, the
Perpendicular Bisector Theorem. You may want to use the plan for proof
given on page 265.
¯
˘
Æ
GIVEN 䉴 CP is the perpendicular bisector of AB .
PROVE 䉴 C is equidistant from A and B.
29.
PROVING THEOREM 5.2 Use the diagram shown to write a two-column
proof of Theorem 5.2, the Converse of the Perpendicular Bisector Theorem.
GIVEN 䉴 C is equidistant from A and B.
C
PROVE 䉴 C is on the perpendicular
Æ
bisector of AB .
Plan for Proof Use the Perpendicular
¯
˘
¯
˘
Postulate to draw CP fi AB . Show that
¤APC £ ¤BPC by the HL Congruence
Æ
Æ
Theorem. Then AP £ BP, so AP = BP.
30.
A
B
H
PROOF Use the diagram shown.
Æ
GIVEN 䉴 GJ is the perpendicular bisector
Æ
of HK.
FOCUS ON
PEOPLE
P
G
M
PROVE 䉴 ¤GHM £ ¤GKM
31.
RE
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THE WRIGHT
BROTHERS
J
K
EARLY AIRCRAFT On many of the earliest airplanes, wires
connected vertical posts to the edges of the wings, which were
wooden frames covered with cloth. Suppose the lengths of the
wires from the top of a post to the edges of the frame are the same
and the distances from the bottom of the post to the ends of the
two wires are the same. What does that tell you about the post
and the section of frame between the ends of the wires?
In Kitty Hawk, North
Carolina, on December 17,
1903, Orville and Wilbur
Wright became the first
people to successfully fly an
engine-driven, heavier-thanair machine.
5.1 Perpendiculars and Bisectors
269
Page 7 of 8
32.
DEVELOPING PROOF Use the diagram to complete the proof of
Theorem 5.4, the Converse of the Angle Bisector Theorem.
A
GIVEN 䉴 D is in the interior of ™ABC and
Æ˘
Æ˘
is equidistant from BA and BC .
D
B
PROVE 䉴 D lies on the angle bisector
of ™ABC.
C
Statements
Reasons
1. D is in the interior of ™ABC.
Æ˘
Æ˘
? from BA and BC .
2. D is 3. ? = ?
Æ˘
Æ
4. DA fi ? , ? fi BC
? 1. 2. Given
3. Definition of equidistant
4. Definition of distance from
a point to a line
5. If 2 lines are fi, then they
form 4 rt. √.
6. Definition of right triangle
? 7. ? 5. ? 6. Æ
Æ
7. BD £ BD
? 8. 9. ™ABD £ ™CBD
Æ˘
10. BD bisects ™ABC and point D
8. HL Congruence Thm.
? 9. ? 10. is on the bisector of ™ABC.
ICE HOCKEY In Exercises 33–35, use the following information.
In the diagram, the goalie is at point G and the puck is at point P.
The goalie’s job is to prevent the puck from entering the goal.
l
33. When the puck is at the other end of the rink,
the goalie is likely to be standing on line l.
Æ
How is l related to AB ?
P
G
34. As an opposing player with the puck skates
toward the goal, the goalie is likely to move
from line l to other places on the ice. What
should be the relationship between
Æ˘
PG and ™APB?
35. How does m™APB change as the puck
gets closer to the goal? Does this change
make it easier or more difficult for the goalie
to defend the goal? Explain.
36.
goal
Chapter 5 Properties of Triangles
goal
line
B
TECHNOLOGY Use geometry software
B
Æ
to construct AB. Find the midpoint C.
Æ
Draw the perpendicular bisector of AB
through C. Construct a point D along the
Æ
Æ
perpendicular bisector and measure DA and DB.
Move D along the perpendicular bisector.
What theorem does this construction
demonstrate?
270
A
C
A
5.4
5.4
D
Page 8 of 8
Test
Preparation
37. MULTI-STEP PROBLEM Use the map
shown and the following information.
A town planner is trying to decide whether
a new household X should be covered by
fire station A, B, or C.
A
a. Trace the map and draw the segments
Æ Æ
X
Æ
AB , BC, and CA.
b. Construct the perpendicular bisectors of
Æ Æ
B
Æ
AB , BC, and CA. Do the perpendicular
bisectors meet at a point?
C
c. The perpendicular bisectors divide the
town into regions. Shade the region
closest to fire station A red. Shade the
region closest to fire station B blue. Shade
the region closest to fire station C gray.
d.
★ Challenge
Writing In an emergency at household X, which fire station should
respond? Explain your choice.
xy USING ALGEBRA Use the graph at
y
the right.
X (4, 8)
38. Use slopes to show that
Æ˘
Æ
Æ
S (3, 5)
Æ˘
WS fi YX and that WT fi YZ .
W (6, 4)
39. Find WS and WT.
EXTRA CHALLENGE
Æ˘
40. Explain how you know that YW
1
bisects ™XYZ.
www.mcdougallittell.com
Y
(2, 2)
x
T (5, 1)
1
Z(8, 0)
MIXED REVIEW
CIRCLES Find the missing measurement for
the circle shown. Use 3.14 as an approximation
for π. (Review 1.7 for 5.2)
41. radius
42. circumference
12 cm
43. area
CALCULATING SLOPE Find the slope of the line that passes through the
given points. (Review 3.6)
44. A(º1, 5), B(º2, 10)
45. C(4, º3), D(º6, 5)
46. E(4, 5), F(9, 5)
47. G(0, 8), H(º7, 0)
48. J(3, 11), K(º10, 12)
49. L(º3, º8), M(8, º8)
xy USING ALGEBRA Find the value of x. (Review 4.1)
50.
xⴗ
51.
(2x ⴙ 6)ⴗ
xⴗ
31ⴗ
40ⴗ
52.
4x ⴗ
70ⴗ
(10x ⴙ 22)ⴗ
5.1 Perpendiculars and Bisectors
271
Page 1 of 7
5.2
Bisectors of a Triangle
What you should learn
GOAL 1 Use properties of
perpendicular bisectors of a
triangle, as applied in
Example 1.
GOAL 2 Use properties of
angle bisectors of a triangle.
Why you should learn it
GOAL 1
In Lesson 5.1, you studied properties of perpendicular
bisectors of segments and angle bisectors. In this lesson,
you will study the special cases in which the segments
and angles being bisected are parts of a triangle.
perpendicular
bisector
A perpendicular bisector of a triangle is a line (or ray
or segment) that is perpendicular to a side of the triangle
at the midpoint of the side.
䉲 To solve real-life
problems, such as finding the
center of a mushroom ring in
Exs. 24–26.
AL LI
ACTIVITY
Developing
Concepts
FE
RE
USING PERPENDICULAR BISECTORS OF A TRIANGLE
Perpendicular Bisectors of a Triangle
1
Cut four large acute scalene triangles out
of paper. Make each one different.
2
Choose one triangle. Fold the triangle
to form the perpendicular bisectors of the
sides. Do the three bisectors intersect at
the same point?
B
A
C
3
Repeat the process for the other three triangles. What do you observe?
Write your observation in the form of a conjecture.
4
Choose one triangle. Label the vertices A, B, and C. Label the point of
Æ Æ
intersection of the perpendicular bisectors as P. Measure AP, BP, and
Æ
CP. What do you observe?
When three or more lines (or rays or segments) intersect in the same point, they
are called concurrent lines (or rays or segments). The point of intersection of
the lines is called the point of concurrency.
The three perpendicular bisectors of a triangle are concurrent. The point of
concurrency can be inside the triangle, on the triangle, or outside the triangle.
P
P
P
acute triangle
272
Chapter 5 Properties of Triangles
right triangle
obtuse triangle
Page 2 of 7
The point of concurrency of the perpendicular bisectors of a triangle is called the
circumcenter of the triangle. In each triangle at the bottom of page 272, the
circumcenter is at P. The circumcenter of a triangle has a special property, as
described in Theorem 5.5. You will use coordinate geometry to illustrate this
theorem in Exercises 29–31. A proof appears on page 835.
THEOREM
THEOREM 5.5
B
Concurrency of Perpendicular
Bisectors of a Triangle
The perpendicular bisectors of a triangle
intersect at a point that is equidistant from
the vertices of the triangle.
P
A
C
PA = PB = PC
The diagram for Theorem 5.5 shows that the circumcenter is the center of the
circle that passes through the vertices of the triangle. The circle is circumscribed
about ¤ABC. Thus, the radius of this circle is the distance from the center to any
of the vertices.
EXAMPLE 1
RE
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Using Perpendicular Bisectors
FACILITIES PLANNING A company plans
to build a distribution center that is
convenient to three of its major clients. The
planners start by roughly locating the three clients
on a sketch and finding the circumcenter of the
triangle formed.
Client
F
Client
E
a. Explain why using the circumcenter as the
Client
G
location of a distribution center would be
convenient for all the clients.
b. Make a sketch of the triangle formed by the
clients. Locate the circumcenter of the triangle.
Tell what segments are congruent.
SOLUTION
F
a. Because the circumcenter is equidistant from
the three vertices, each client would be equally
close to the distribution center.
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
E
D
b. Label the vertices of the triangle as E, F, and G.
Draw the perpendicular bisectors. Label their
intersection as D.
䉴
G
By Theorem 5.5, DE = DF = DG.
5.2 Bisectors of a Triangle
273
Page 3 of 7
GOAL 2
USING ANGLE BISECTORS OF A TRIANGLE
An angle bisector of a triangle is a bisector of an
angle of the triangle. The three angle bisectors are
concurrent. The point of concurrency of the angle
bisectors is called the incenter of the triangle,
and it always lies inside the triangle. The incenter
has a special property that is described below in
Theorem 5.6. Exercise 22 asks you to write a proof
of this theorem.
P
THEOREM
THEOREM 5.6
Concurrency of Angle
Bisectors of a Triangle
B
D
The angle bisectors of a triangle intersect
at a point that is equidistant from the
sides of the triangle.
F
P
PD = PE = PF
E
A
C
The diagram for Theorem 5.6 shows that the incenter is the center of the circle
that touches each side of the triangle once. The circle is inscribed within ¤ABC.
Thus, the radius of this circle is the distance from the center to any of the sides.
EXAMPLE 2
Logical
Reasoning
Using Angle Bisectors
The angle bisectors of ¤MNP meet at point L.
M
S
P
17
a. What segments are congruent?
b. Find LQ and LR.
15
R
L
q
SOLUTION
a. By Theorem 5.6, the three angle bisectors of a
triangle intersect at a point that is equidistant from
Æ
Æ Æ
the sides of the triangle. So, LR £ LQ £ LS.
STUDENT HELP
b. Use the Pythagorean Theorem to find LQ in ¤LQM.
(LQ)2 + (MQ)2 = (LM)2
Look Back
For help with the
Pythagorean Theorem,
see p. 20.
(LQ)2 + 152 = 172
Substitute.
(LQ)2 + 225 = 289
Multiply.
2
(LQ) = 64
LQ = 8
䉴
274
Subtract 225 from each side.
Find the positive square root.
Æ
Æ
So, LQ = 8 units. Because LR £ LQ, LR = 8 units.
Chapter 5 Properties of Triangles
N
Page 4 of 7
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Skill Check
✓
?.
1. If three or more lines intersect at the same point, the lines are 2. Think of something about the words incenter and circumcenter that you can
use to remember which special parts of a triangle meet at each point.
Use the diagram and the given information to find the indicated measure.
3. The perpendicular bisectors of
4. The angle bisectors of ¤XYZ meet
¤ABC meet at point G. Find GC.
A
at point M. Find MK.
X
E
7
D
L
M
12
Z
8
5
K
G
5
J
2
C
F
Y
B
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 811.
CONSTRUCTION Draw a large example of the given type of triangle.
Construct perpendicular bisectors of the sides. (See page 264.) For the type
of triangle, do the bisectors intersect inside, on, or outside the triangle?
5. obtuse triangle
6. acute triangle
7. right triangle
DRAWING CONCLUSIONS Draw a large ¤ABC.
8. Construct the angle bisectors of ¤ABC. Label the point where the angle
bisectors meet as D.
9. Construct perpendicular segments from D to each of the sides of the triangle.
Measure each segment. What do you notice? Which theorem have you just
confirmed?
LOGICAL REASONING Use the results of Exercises 5–9 to complete the
statement using always, sometimes, or never.
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 5–7,
10–13, 14, 17, 20, 21
Example 2: Exs. 8, 9,
10–13, 15, 16, 22
? passes through the midpoint of a
10. A perpendicular bisector of a triangle side of the triangle.
? intersect at a single point.
11. The angle bisectors of a triangle ? meet at a point outside the triangle.
12. The angle bisectors of a triangle ? lies outside the triangle.
13. The circumcenter of a triangle 5.2 Bisectors of a Triangle
275
Page 5 of 7
BISECTORS In each case, find the indicated measure.
14. The perpendicular bisectors of
15. The angle bisectors of ¤XYZ meet
¤RST meet at point D. Find DR.
at point W. Find WB.
S
Z
W
4.68
9
B
A
Y
20
75
D
C
R
X
T
16
16. The angle bisectors of ¤GHJ
17. The perpendicular bisectors of ¤MNP
meet at point K. Find KB.
meet at point Q. Find QN.
J
N
M
B
4
5
q
H
K
C
7
48
A
P
G
ERROR ANALYSIS Explain why the student’s conclusion is false. Then state
a correct conclusion that can be deduced from the diagram.
18.
19.
B
J
Q
K
E
F
P
M
D
A
G
C
N
L
MQ = MN
DE = DG
LOGICAL REASONING In Exercises 20 and 21, use the following
information and map.
Your family is considering moving to a new
home. The diagram shows the locations of
where your parents work and where you go to
school. The locations form a triangle.
school
factory
20. In the diagram, how could you find a
point that is equidistant from each
location? Explain your answer.
21. Make a sketch of the situation. Find the
best location for the new home.
276
Chapter 5 Properties of Triangles
office
Page 6 of 7
22.
DEVELOPING PROOF Complete the proof of Theorem 5.6, the
Concurrency of Angle Bisectors.
GIVEN 䉴 ¤ABC, the bisectors of ™A, ™B, and
Æ
Æ Æ
Æ Æ
C
Æ
™C, DE fi AB, DF fi BC, DG fi CA
PROVE 䉴 The angle bisectors intersect at a point
Æ Æ
Æ
F
that is equidistant from AB, BC, and CA.
G
Plan for Proof Show that D, the point of
intersection of the bisectors of ™A and ™B, also
lies on the bisector of ™C. Then show that D is
equidistant from the sides of the triangle.
Statements
D
A
B
E
Reasons
1. ¤ABC, the bisectors of ™A,
Æ
Æ
1. Given
™B, and ™C, DE fi AB,
Æ
Æ Æ
Æ
DF fi BC, DG fi CA
Æ˘
2. ? = DG
?
2. AD bisects ™BAC, so D is from the sides of ™BAC.
3. DE = DF
?
3. 4. DF = DG
?
4. ? of ™C.
5. D is on the 5. Converse of the Angle Bisector
Theorem
?
6. 23.
?
6. Givens and Steps Writing Joannie thinks that the midpoint
R
of the hypotenuse of a right triangle is
equidistant from the vertices of the triangle.
Explain how she could use perpendicular
bisectors to verify her conjecture.
q
T
FOCUS ON
APPLICATIONS
SCIENCE
CONNECTION
S
In Exercises 24–26, use the following information.
A mycelium fungus grows underground in all directions from a central point.
Under certain conditions, mushrooms sprout up in a ring at the edge. The radius
of the mushroom ring is an indication of the mycelium’s age.
24. Suppose three mushrooms in a mushroom ring
RE
FE
L
AL I
MUSHROOMS live
for only a few days.
As the mycelium spreads
outward, new mushroom
rings are formed. A
mushroom ring in France
is almost half a mile in
diameter and is about
700 years old.
y
are located as shown. Make a large copy of the
diagram and draw ¤ABC. Each unit on your
coordinate grid should represent 1 foot.
A(2, 5)
B(6, 3)
25. Draw perpendicular bisectors on your diagram
1
to find the center of the mushroom ring.
Estimate the radius of the ring.
C(4, 1)
1
x
26. Suppose the radius of the mycelium increases
at a rate of about 8 inches per year. Estimate
its age.
5.2 Bisectors of a Triangle
277
Page 7 of 7
Test
Preparation
MULTIPLE CHOICE Choose the correct answer from the list given.
Æ
Æ
27. AD and CD are angle bisectors of ¤ABC
B
and m™ABC = 100°. Find m™ADC.
100ⴗ
A
¡
D
¡
80°
120°
B
¡
E
¡
C
¡
90°
D
100°
140°
A
28. The perpendicular bisectors of ¤XYZ
X
intersect at point W, WT = 12, and
WZ = 13. Find XY.
A
¡
D
¡
B
¡
E
¡
5
12
C
¡
8
C
W
12
13
T
10
13
Z
Y
★ Challenge
xy USING ALGEBRA Use the graph of ¤ABC to illustrate Theorem 5.5, the
Concurrency of Perpendicular Bisectors.
29. Find the midpoint of each side of ¤ABC.
y
Use the midpoints to find the equations
of the perpendicular bisectors of ¤ABC.
B (12, 6)
30. Using your equations from Exercise 29,
find the intersection of two of the lines.
Show that the point is on the third line.
EXTRA CHALLENGE
www.mcdougallittell.com
2
A (0, 0)
31. Show that the point in Exercise 30 is
8
x
C (18, 0)
equidistant from the vertices of ¤ABC.
MIXED REVIEW
FINDING AREAS Find the area of the triangle described. (Review 1.7 for 5.3)
32. base = 9, height = 5
33. base = 22, height = 7
WRITING EQUATIONS The line with the given equation is perpendicular to
line j at point P. Write an equation of line j. (Review 3.7)
34. y = 3x º 2, P(1, 4)
35. y = º2x + 5, P(7, 6)
2
36. y = ºᎏᎏx º 1, P(2, 8)
3
10
37. y = ᎏᎏx + 3, P(º2, º9)
11
LOGICAL REASONING Decide whether enough information is given to
prove that the triangles are congruent. If there is enough information, tell
which congruence postulate or theorem you would use. (Review 4.3, 4.4, and 4.6)
38. A
B
40. P
39. F
M
8
8
K
5
J
C
E
278
Chapter 5 Properties of Triangles
5
D
H
10
G
N
10
L
Page 1 of 7
5.3
Medians and Altitudes
of a Triangle
What you should learn
GOAL 1 Use properties of
medians of a triangle.
GOAL 2 Use properties of
altitudes of a triangle.
Why you should learn it
RE
FE
䉲 To solve real-life
problems, such as locating
points in a triangle used
to measure a person’s
heart fitness as in
Exs. 30–33.
AL LI
GOAL 1
USING MEDIANS OF A TRIANGLE
In Lesson 5.2, you studied two special types of
segments of a triangle: perpendicular bisectors of
the sides and angle bisectors. In this lesson, you
will study two other special types of segments of
a triangle: medians and altitudes.
A
median
A median of a triangle is a segment whose
endpoints are a vertex of the triangle and the
midpoint of the opposite side. For instance, in
¤ABC shown at the right, D is the midpoint of
Æ
Æ
side BC. So, AD is a median of the triangle.
B
D
C
The three medians of a triangle are concurrent. The point of concurrency is called
the centroid of the triangle. The centroid, labeled P in the diagrams below, is
always inside the triangle.
P
acute triangle
P
P
right triangle
obtuse triangle
The medians of a triangle have a special concurrency property, as described in
Theorem 5.7. Exercises 13–16 ask you to use paper folding to demonstrate the
relationships in this theorem. A proof appears on pages 836–837.
THEOREM
THEOREM 5.7
Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point that is two thirds of the
distance from each vertex to the midpoint of the opposite side.
If P is the centroid of ¤ABC, then
2
2
2
AP = ᎏᎏ AD, BP = ᎏᎏ BF, and CP = ᎏᎏCE.
3
3
3
B
D
P
C
E
F
A
The centroid of a triangle can be used as its balancing point, as shown on the
next page.
5.3 Medians and Altitudes of a Triangle
279
Page 2 of 7
FOCUS ON
APPLICATIONS
1990
RE
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centroid
A triangular model of uniform thickness and
density will balance at the centroid of the
triangle. For instance, in the diagram shown
at the right, the triangular model will balance
if the tip of a pencil is placed at its centroid.
1890 1790
CENTER OF
POPULATION
Suppose the location of
each person counted in a
census is identified by a
weight placed on a flat,
weightless map of the United
States. The map would
balance at a point that is the
center of the population.
This center has been moving
westward over time.
EXAMPLE 1
Using the Centroid of a Triangle
P is the centroid of ¤QRS shown below and PT = 5. Find RT and RP.
SOLUTION
2
3
Because P is the centroid, RP = ᎏᎏ RT.
R
1
3
Then PT = RT º RP = ᎏᎏ RT.
1
3
Substituting 5 for PT, 5 = ᎏᎏ RT, so RT = 15.
P
q
T
S
2
2
Then RP = ᎏᎏ RT = ᎏᎏ(15) = 10.
3
3
䉴
So, RP = 10 and RT = 15.
EXAMPLE 2
Finding the Centroid of a Triangle
Find the coordinates of the centroid of ¤JKL.
y
J (7, 10)
SOLUTION
N
You know that the centroid is two thirds
of the distance from each vertex to the
midpoint of the opposite side.
Æ
Choose the median KN. Find the
M
Æ
coordinates of N, the midpoint of JL .
The coordinates of N are
6 + 10
10 16
ᎏ = 冉ᎏᎏ, ᎏ2ᎏ冊 = (5, 8).
冉ᎏ3 +2ᎏ7 , ᎏ
2 冊
2
P
L(3, 6)
K (5, 2)
1
1
x
Find the distance from vertex K to midpoint N. The distance from K(5, 2) to
N(5, 8) is 8 º 2, or 6 units.
2
Determine the coordinates of the centroid, which is ᎏᎏ • 6, or 4 units up from
3
Æ
vertex K along the median KN.
䉴
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
280
The coordinates of centroid P are (5, 2 + 4), or (5, 6).
..........
Exercises 21–23 ask you to use the Distance Formula to confirm that the distance
from vertex J to the centroid P in Example 2 is two thirds of the distance from J
to M, the midpoint of the opposite side.
Chapter 5 Properties of Triangles
Page 3 of 7
GOAL 2
USING ALTITUDES OF A TRIANGLE
An altitude of a triangle is the perpendicular segment from a vertex to the
opposite side or to the line that contains the opposite side. An altitude can lie
inside, on, or outside the triangle.
Every triangle has three altitudes. The lines containing the altitudes are
concurrent and intersect at a point called the orthocenter of the triangle.
EXAMPLE 3
Logical
Reasoning
Drawing Altitudes and Orthocenters
Where is the orthocenter located in each type of triangle?
a. Acute triangle
b. Right triangle
c. Obtuse triangle
SOLUTION
Draw an example of each type of triangle and locate its orthocenter.
K
B
E
A
F
Y
W
D
G
P
Z
J
M
L
q
X
C
R
a. ¤ABC is an acute triangle. The three altitudes intersect at G, a point inside
the triangle.
Æ
Æ
b. ¤KLM is a right triangle. The two legs, LM and KM, are also altitudes. They
intersect at the triangle’s right angle. This implies that the orthocenter is on
the triangle at M, the vertex of the right angle of the triangle.
c. ¤YPR is an obtuse triangle. The three lines that contain the altitudes intersect
at W, a point that is outside the triangle.
THEOREM
THEOREM 5.8
Concurrency of Altitudes of a Triangle
The lines containing the altitudes
of a triangle are concurrent.
Æ Æ
H
F
Æ
If AE , BF , and CD are the altitudes of
¯
˘¯
˘
¯
˘
¤ABC, then the lines AE, BF, and CD
intersect at some point H.
B
A
E
D
C
Exercises 24–26 ask you to use construction to verify Theorem 5.8. A proof
appears on page 838.
5.3 Medians and Altitudes of a Triangle
281
Page 4 of 7
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Skill Check
✓
?㛭㛭㛭 intersect.
1. The centroid of a triangle is the point where the three 㛭㛭㛭㛭㛭
2. In Example 3 on page 281, explain why the two legs of the right triangle in
part (b) are also altitudes of the triangle.
Use the diagram shown and the given information to decide in each case
Æ
whether EG is a perpendicular bisector, an angle bisector, a median, or an
altitude of ¤DEF.
Æ
Æ
3. DG £ FG
Æ
E
Æ
4. EG fi DF
5. ™DEG £ ™FEG
Æ
Æ
Æ
Æ
6. EG fi DF and DG £ FG
7. ¤DGE £ ¤FGE
D
G
F
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 811.
USING MEDIANS OF A TRIANGLE In Exercises 8–12, use the figure below and
the given information.
Æ
Æ
E
P is the centroid of ¤DEF, EH fi DF,
DH = 9, DG = 7.5, EP = 8, and DE = FE.
8
Æ
8. Find the length of FH.
G
Æ
Æ
J
7.5
9. Find the length of EH.
10. Find the length of PH.
P
D
9
H
F
11. Find the perimeter of ¤DEF.
12.
EP
2
LOGICAL REASONING In the diagram of ¤DEF above, ᎏᎏ = ᎏᎏ.
EH
3
PH
PH
Find ᎏᎏ and ᎏᎏ.
EH
EP
PAPER FOLDING Cut out a large acute, right, or obtuse triangle. Label the
vertices. Follow the steps in Exercises 13–16 to verify Theorem 5.7.
13. Fold the sides to locate the midpoint of each side.
A
Label the midpoints.
14. Fold to form the median from each vertex to the
STUDENT HELP
midpoint of the opposite side.
HOMEWORK HELP
15. Did your medians meet at about the same
Example 1: Exs. 8–11,
13–16
Example 2: Exs. 17–23
Example 3: Exs. 24–26
point? If so, label this centroid point.
282
16. Verify that the distance from the centroid to a
vertex is two thirds of the distance from that
vertex to the midpoint of the opposite side.
Chapter 5 Properties of Triangles
L
C
M
N
B
Page 5 of 7
xy USING ALGEBRA Use the graph shown.
y
17. Find the coordinates of Q, the
Æ
midpoint of MN.
P (5, 6)
Æ
18. Find the length of the median PQ.
19. Find the coordinates of the
centroid. Label this point as T.
R
2
N (11, 2)
20. Find the coordinates of R, the
x
10
œ
M (⫺1, ⫺2)
Æ
midpoint of MP. Show that the
NT
NR
2
3
quotient ᎏᎏ is ᎏᎏ.
xy USING ALGEBRA Refer back to Example 2 on page 280.
Æ
21. Find the coordinates of M, the midpoint of KL.
Æ
Æ
22. Use the Distance Formula to find the lengths of JP and JM.
2
23. Verify that JP = ᎏᎏJM.
3
STUDENT HELP
Look Back
To construct an altitude,
use the construction of a
perpendicular to a line
through a point not on the
line, as shown on p. 130.
CONSTRUCTION Draw and label a large scalene triangle of the given
type and construct the altitudes. Verify Theorem 5.8 by showing that the
lines containing the altitudes are concurrent, and label the orthocenter.
24. an acute ¤ABC
25. a right ¤EFG with
26. an obtuse ¤KLM
right angle at G
TECHNOLOGY Use geometry software to draw a triangle. Label the
vertices as A, B, and C.
27. Construct the altitudes of ¤ABC by drawing perpendicular lines through
Æ Æ
Æ
each side to the opposite vertex. Label them AD, BE, and CF.
Æ
Æ
Æ
Æ
28. Find and label G and H, the intersections of AD and BE and of BE and CF.
29. Prove that the altitudes are concurrent by showing that GH = 0.
FOCUS ON
CAREERS
ELECTROCARDIOGRAPH In Exercises 30–33, use the following
information about electrocardiographs.
The equilateral triangle ¤BCD is used to plot electrocardiograph readings.
Consider a person who has a left shoulder reading (S) of º1, a right shoulder
reading (R) of 2, and a left leg reading (L) of 3.
Right shoulder
0
2
⫺4 ⫺2
30. On a large copy of ¤BCD, plot the
reading to form the vertices of ¤SRL.
(This triangle is an Einthoven’s
Triangle, named for the inventor of the
electrocardiograph.)
RE
FE
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CARDIOLOGY
TECHNICIAN
INT
Technicians use equipment
like electrocardiographs to
test, monitor, and evaluate
heart function.
NE
ER T
CAREER LINK
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31. Construct the circumcenter M of ¤SRL.
32. Construct the centroid P of ¤SRL.
Æ
Draw line r through P parallel to BC.
33. Estimate the measure of the acute angle
Æ
B
4
⫺4
⫺4
⫺2
⫺2
0
Left
shoulder
C
0
2
2
4
Left
leg
4
D
between line r and MP. Cardiologists
call this the angle of a person’s heart.
5.3 Medians and Altitudes of a Triangle
283
Page 6 of 7
Test
Preparation
34. MULTI-STEP PROBLEM Recall the formula for the area of a triangle,
1
A = ᎏᎏbh, where b is the length of the base and h is the height. The height of
2
a triangle is the length of an altitude.
a. Make a sketch of ¤ABC. Find CD, the height of
Æ
the triangle (the length of the altitude to side AB).
C
b. Use CD and AB to find the area of ¤ABC.
E
15
Æ
c. Draw BE, the altitude to the line containing
Æ
side AC.
D
12
A
d. Use the results of part (b) to find
8
Æ
the length of BE.
e.
Writing Write a formula for the length of an altitude in terms of the base
and the area of the triangle. Explain.
★ Challenge
SPECIAL TRIANGLES Use the diagram at the right.
35. GIVEN 䉴 ¤ABC is isosceles.
Æ
A
Æ
BD is a median to base AC.
Æ
B
PROVE 䉴 BD is also an altitude.
D
36. Are the medians to the legs of an isosceles
C
triangle also altitudes? Explain your reasoning.
37. Are the medians of an equilateral triangle also altitudes? Are they contained
in the angle bisectors? Are they contained in the perpendicular bisectors?
EXTRA CHALLENGE
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38.
LOGICAL REASONING In a proof, if you are given a median of an
equilateral triangle, what else can you conclude about the segment?
MIXED REVIEW
xy USING ALGEBRA Write an equation of the line that passes through
point P and is parallel to the line with the given equation. (Review 3.6 for 5.4)
39. P(1, 7), y = ºx + 3
40. P(º3, º8), y = º2x º 3
1
42. P(4, º2), y = ºᎏᎏ x º 1
2
41. P(4, º9), y = 3x + 5
DEVELOPING PROOF In Exercises 43 and 44, state the third congruence
that must be given to prove that ¤DEF £ ¤GHJ using the indicated
postulate or theorem. (Review 4.4)
E
Æ
Æ
43. GIVEN 䉴 ™D £ ™G, DF £ GJ
H
AAS Congruence Theorem
Æ
Æ
44. GIVEN 䉴 ™E £ ™H, EF £ HJ
ASA Congruence Postulate
D
F
G
45. USING THE DISTANCE FORMULA Place a right triangle with legs
of length 9 units and 13 units in a coordinate plane and use the
Distance Formula to find the length of the hypotenuse. (Review 4.7)
284
Chapter 5 Properties of Triangles
J
B
Page 7 of 7
QUIZ 1
Self-Test for Lessons 5.1– 5.3
Use the diagram shown and the given
information. (Lesson 5.1)
Æ
y ⴙ 24 L
J
K
Æ
3y
HJ is the perpendicular bisector of KL .
Æ˘
HJ bisects ™KHL.
4x ⴙ 9
3x ⴙ 25
1. Find the value of x.
H
2. Find the value of y.
T
In the diagram shown, the perpendicular
bisectors of ¤RST meet at V. (Lesson 5.2)
6
Æ
8
3. Find the length of VT .
V
Æ
4. What is the length of VS ? Explain.
5.
R
BUILDING A MOBILE Suppose you
want to attach the items in a mobile so
that they hang horizontally. You would
want to find the balancing point of
each item. For the triangular metal plate
shown, describe where the balancing
point would be located. (Lesson 5.3)
S
A
E
C
F
G
D
B
Æ Æ
Æ
INT
AD , BE , and CF are medians. CF = 12 in.
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Optimization
THEN
NOW
APPLICATION LINK
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THROUGHOUT HISTORY, people have faced problems involving minimizing resources
or maximizing output, a process called optimization. The use of mathematics in
solving these types of problems has increased greatly since World War II, when
mathematicians found the optimal shape for naval convoys to avoid enemy fire.
M
TODAY, with the help of computers, optimization techniques are used in
many industries, including manufacturing, economics, and architecture.
1. Your house is located at point H in the diagram. You need to do errands
P
H
at the post office (P), the market (M), and the library (L). In what order
should you do your errands to minimize the distance traveled?
L
2. Look back at Exercise 34 on page 270. Explain why the goalie’s position
on the angle bisector optimizes the chances of blocking a scoring shot.
WWII naval convoy
Thomas Hales proves Kepler’s
cannonball conjecture.
1942
1611
Johannes Kepler proposes
the optimal way to stack
cannonballs.
1972
This Olympic stadium
roof uses a minimum
of materials.
1997
5.3 Medians and Altitudes of a Triangle
285
Page 1 of 7
5.4
Midsegment Theorem
What you should learn
GOAL 1 Identify the
midsegments of a triangle.
GOAL 2 Use properties of
midsegments of a triangle.
Why you should learn it
RE
USING MIDSEGMENTS OF A TRIANGLE
In Lessons 5.2 and 5.3, you studied four special types of segments of a triangle:
perpendicular bisectors, angle bisectors, medians, and altitudes. Another special
type of segment is called a midsegment. A midsegment of a triangle is a
segment that connects the midpoints of two sides of a triangle.
You can form the three midsegments of a triangle by tracing the triangle on
paper, cutting it out, and folding it, as shown below.
FE
䉲 To solve real-life
problems involving
midsegments, as applied in
Exs. 32 and 35.
AL LI
GOAL 1
1
Fold one vertex
onto another to
find one
midpoint.
2
Repeat the
process to find
the other two
midpoints.
3
Fold a
segment that
contains two of
the midpoints.
4
Fold the
remaining two
midsegments
of the triangle.
The midsegments and sides of a triangle have a special relationship, as shown in
Example 1 and Theorem 5.9 on the next page.
The roof of the Cowles
Conservatory in
Minneapolis, Minnesota,
shows the midsegments
of a triangle.
EXAMPLE 1
Using Midsegments
Æ
Show that the midsegment MN is parallel to
Æ
side JK and is half as long.
y
K (4, 5)
J(⫺2, 3)
SOLUTION
N
Use the Midpoint Formula to find the
coordinates of M and N.
1
冉
冊
4 + 6 5 + (º1)
N = 冉ᎏᎏ, ᎏᎏ冊 = (5, 2)
2
2
º2 + 6 3 + (º1)
M = ᎏᎏ, ᎏᎏ = (2, 1)
2
2
Æ
M
1
x
L(6, ⫺1)
Æ
Next, find the slopes of JK and MN.
Æ
5º3
4 º (º2)
2
6
1
3
Æ
Slope of JK = ᎏᎏ = ᎏᎏ = ᎏᎏ
䉴
Æ
2º1
5º2
1
3
Slope of MN = ᎏᎏ = ᎏᎏ
Æ
Because their slopes are equal, JK and MN are parallel. You can use the
Distance Formula to show that MN = 兹1苶0苶 and JK = 兹4苶0苶 = 2兹1苶0苶. So,
Æ
Æ
MN is half as long as JK .
5.4 Midsegment Theorem
287
Page 2 of 7
THEOREM
Midsegment Theorem
THEOREM 5.9
C
The segment connecting the midpoints of
two sides of a triangle is parallel to the
third side and is half as long.
Æ
Æ
E
D
1
2
DE ∞ AB and DE = ᎏᎏAB
B
A
Using the Midsegment Theorem
EXAMPLE 2
Æ
Æ
UW and VW are midsegments of ¤RST. Find UW and RT.
R
SOLUTION
U
1
1
UW = ᎏᎏ(RS) = ᎏᎏ(12) = 6
2
2
12 V
8
T
W
RT = 2(VW) = 2(8) = 16
..........
S
A coordinate proof of Theorem 5.9 for one midsegment of a triangle is given
below. Exercises 23–25 ask for proofs about the other two midsegments. To set
up a coordinate proof, remember to place the figure in a convenient location.
Proving Theorem 5.9
EXAMPLE 3
Proof
y
Write a coordinate proof of the Midsegment Theorem.
C(2a, 2b)
SOLUTION
D
E
Place points A, B, and C in convenient locations in a
x
coordinate plane, as shown. Use the Midpoint Formula
to find the coordinates of the midpoints D and E.
冉 2a 2+ 0 2b 2+ 0 冊
D = ᎏᎏ, ᎏᎏ = (a, b)
STUDENT HELP
Study Tip
In Example 3, it is
convenient to locate a
vertex at (0, 0) and it also
helps to make one side
horizontal. To use the
Midpoint Formula, it is
helpful for the coordinates
to be multiples of 2.
288
A (0, 0)
B (2c, 0)
冉 2a +2 2c 2b 2+ 0 冊
E = ᎏᎏ, ᎏᎏ = (a + c, b)
Æ
Find the slope of midsegment DE. Points D and E have the same y-coordinates,
Æ
so the slope of DE is zero.
䉴
Æ
Æ
Æ
AB also has a slope of zero, so the slopes are equal and DE and AB are parallel.
Æ
Æ
Calculate the lengths of DE and AB. The segments are both horizontal, so their
lengths are given by the absolute values of the differences of their x-coordinates.
AB = |2c º 0| = 2c
䉴
Æ
DE = |a + c º a| = c
Æ
The length of DE is half the length of AB.
Chapter 5 Properties of Triangles
Page 3 of 7
GOAL 2
USING PROPERTIES OF MIDSEGMENTS
Suppose you are given only the three midpoints of the sides of a triangle. Is it
possible to draw the original triangle? Example 4 shows one method.
xy
Using
Algebra
Using Midpoints to Draw a Triangle
EXAMPLE 4
The midpoints of the sides of a triangle are L(4, 2), M(2, 3), and N(5, 4). What
are the coordinates of the vertices of the triangle?
slope ⫽
SOLUTION
y
4⫺3
5⫺2
⫽
Plot the midpoints in a coordinate plane.
Connect these midpoints to form the
Æ Æ
Æ
midsegments LN, MN , and ML.
Find the slopes of the midsegments.
Use the slope formula as shown.
1
3
N
M
slope ⫽
3⫺2
2⫺4
⫽ ⫺ 12
L
1
1
Each midsegment contains two of the
unknown triangle’s midpoints and is
parallel to the side that contains the third
midpoint. So, you know a point on each side
of the triangle and the slope of each side.
x
slope ⫽
y
4⫺2
5⫺4
⫽2
A
N
M
B
Draw the lines that contain the three sides.
1
䉴
The lines intersect at A(3, 5), B(7, 3), and
C(1, 1), which are the vertices of the triangle.
..........
L
C
1
x
The perimeter of the triangle formed by the three midsegments of a triangle is
half the perimeter of the original triangle, as shown in Example 5.
Perimeter of Midsegment Triangle
EXAMPLE 5
FOCUS ON
APPLICATIONS
Æ Æ
Æ
ORIGAMI DE, EF, and DF are midsegments
A
10 cm
in ¤ABC. Find the perimeter of ¤DEF.
B
E
SOLUTION The lengths of the midsegments
are half the lengths of the sides of ¤ABC.
1
2
1
2
1
2
1
2
1
2
1
2
DF = ᎏᎏAB = ᎏᎏ(10) = 5
10 cm
D
F
14.2 cm
EF = ᎏᎏAC = ᎏᎏ(10) = 5
RE
FE
L
AL I
ORIGAMI is an
ancient method
of paper folding. The
pattern of folds for a
number of objects, such
as the flower shown,
involve midsegments.
ED = ᎏᎏBC = ᎏᎏ(14.2) = 7.1
C
Crease pattern of origami flower
䉴
The perimeter of ¤DEF is 5 + 5 + 7.1, or 17.1. The perimeter of ¤ABC is
10 + 10 + 14.2, or 34.2, so the perimeter of the triangle formed by the
midsegments is half the perimeter of the original triangle.
5.4 Midsegment Theorem
289
Page 4 of 7
GUIDED PRACTICE
Vocabulary Check
Concept Check
Skill Check
✓
1. In ¤ABC, if M is the midpoint of AB, N is the midpoint of AC, and P is the
✓
2. In Example 3 on page 288, why was it convenient to position one of the sides
✓
Æ
Æ
Æ Æ
Æ
Æ
? of ¤ABC.
midpoint of BC, then MN, NP, and PN are of the triangle along the x-axis?
Æ Æ
Æ
In Exercises 3–9, GH , HJ , and JG are
midsegments of ¤DEF.
Æ
D
Æ
?
3. JH ∞ ? ∞ DE
4. ?
5. EF = ?
6. GH = ?
7. DF = ?
8. JH = 24
J
E
10.6
8
H
G
F
9. Find the perimeter of ¤GHJ.
WALKWAYS The triangle below shows a section of walkways on a
college campus.
Æ
y
10. The midsegment AB represents a new
walkway that is to be constructed on the
campus. What are the coordinates of
points A and B?
œ (2, 8)
B
A
11. Each unit in the coordinate plane represents
R (10, 4)
2
10 yards. Use the Distance Formula to find
the length of the new walkway.
O (0, 0)
6
x
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 812.
COMPLETE THE STATEMENT In Exercises 12–19, use ¤ABC, where L, M,
and N are midpoints of the sides.
12. LM
Æ
?
∞ Æ
?
∞ 13. AB
B
L
? .
14. If AC = 20, then LN = N
? .
15. If MN = 7, then AB = ? .
16. If NC = 9, then LM = A
M
C
? .
17. xy USING ALGEBRA If LM = 3x + 7 and BC = 7x + 6, then LM = STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 21, 22
Example 2: Exs. 12–16
Example 3: Exs. 23–25
Example 4: Exs. 26, 27
Example 5: Exs. 28, 29
290
? .
18. xy USING ALGEBRA If MN = x º 1 and AB = 6x º 18, then AB = 19.
LOGICAL REASONING Which angles in the diagram are congruent?
Explain your reasoning.
20.
CONSTRUCTION Use a straightedge to draw a triangle. Then use the
straightedge and a compass to construct the three midsegments of the triangle.
Chapter 5 Properties of Triangles
Page 5 of 7
xy USING ALGEBRA Use the diagram.
y
21. Find the coordinates of the endpoints
C (10, 6)
6
of each midsegment of ¤ABC.
F
A (0, 2)
22. Use slope and the Distance Formula to
E
verify that the Midsegment Theorem is
Æ
true for DF.
10
D
x
B (5, ⫺2)
xy USING ALGEBRA Copy the diagram in Example 3 on page 288 to
complete the proof of Theorem 5.9, the Midsegment Theorem.
Æ
23. Locate the midpoint of AB and label it F. What are the coordinates of F?
Æ
Æ
Draw midsegments DF and EF.
Æ
Æ
Æ
Æ
24. Use slopes to show that DF ∞ CB and EF ∞ CA.
25. Use the Distance Formula to find DF, EF, CB, and CA. Verify that
1
1
DF = ᎏᎏCB and EF = ᎏᎏCA.
2
2
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with
Exs. 26 and 27.
xy USING ALGEBRA In Exercises 26 and 27, you are given the midpoints of
the sides of a triangle. Find the coordinates of the vertices of the triangle.
26. L(1, 3), M(5, 9), N(4, 4)
27. L(7, 1), M(9, 6), N(5, 4)
FINDING PERIMETER In Exercises 28 and 29, use the diagram shown.
28. Given CD = 14, GF = 8,
and GC = 5, find the perimeter
of ¤BCD.
29. Given PQ = 20, SU = 12,
and QU = 9, find the perimeter
of ¤STU.
T
P
C
G
R
F
U
S
B
E
q
D
FOCUS ON
APPLICATIONS
30.
TECHNOLOGY Use geometry software to draw any ¤ABC. Construct
Æ Æ
Æ
the midpoints of AB, BC, and CA. Label them as D, E, and F. Construct
Æ Æ
Æ
the midpoints of DE, EF , and FD. Label them as G, H, and I. What is the
relationship between the perimeters of ¤ABC and ¤GHI?
31. FRACTALS The design below, which approximates a fractal, is created with
L
AL I
RE
INT
FE
FRACTALS are
shapes that look the
same at many levels of
magnification. Take a small
part of the image above and
you will see that it looks
about the same as the
whole image.
midsegments. Beginning with any triangle, shade the triangle formed by the
three midsegments. Continue the process for each unshaded triangle.
Suppose the perimeter of the original triangle is 1. What is the perimeter of
the triangle that is shaded in Stage 1? What is the total perimeter of all the
triangles that are shaded in Stage 2? in Stage 3?
NE
ER T
APPLICATION LINK
www.mcdougallittell.com
Stage 0
Stage 1
Stage 2
Stage 3
5.4 Midsegment Theorem
291
Page 6 of 7
32.
33.
PORCH SWING You are assembling
the frame for a porch swing. The
horizontal crossbars in the kit you
purchased are each 30 inches long. You
attach the crossbars at the midpoints of
the legs. At each end of the frame, how
far apart will the bottoms of the legs be
when the frame is assembled? Explain.
crossbar
?
WRITING A PROOF Write a paragraph proof using the diagram shown
and the given information.
Æ Æ
Æ
GIVEN 䉴 ¤ABC with midsegments DE , EF , and FD
A
PROVE 䉴 ¤ADE £ ¤DBF
D
Plan for Proof Use the SAS Congruence
Æ
E
Æ
Postulate. Show that AD £ DB. Show that
Æ
Æ
1
because DE = BF = ᎏᎏBC, then DE £ BF.
2
B
F
C
Use parallel lines to show that ™ADE £ ™ABC.
Test
Preparation
STUDENT HELP
Skills Review
For help with writing
an equation of a line,
see page 795.
292
34.
WRITING A PLAN Using the information from Exercise 33, write a plan
for a proof showing how you could use the SSS Congruence Postulate to
prove that ¤ADE £ ¤DBF.
35.
A-FRAME HOUSE In the A-frame
house shown, the floor of the second level,
Æ
Æ
labeled PQ, is closer to the first floor, RS ,
Æ
Æ
than midsegment MN is. If RS is 24 feet
Æ
long, can PQ be 10 feet long? 12 feet long?
14 feet long? 24 feet long? Explain.
36. MULTI-STEP PROBLEM The diagram below shows the points D(2, 4), E(3, 2),
and F(4, 5), which are midpoints of the sides of ¤ABC. The directions below
show how to use equations of lines to reconstruct the original ¤ABC.
a. Plot D, E, and F in a coordinate plane.
y
b. Find the slope m1 of one midsegment, say
Æ
DE.
C
Æ
c. The line containing side CB will have the
F (4, 5)
Æ
Æ
same slope as DE. Because CB contains
D(2, 4)
¯
˘
F(4, 5), an equation of CB in point-slope
B
form is y º 5 = m1(x º 4). Write an
E (3, 2)
¯
˘
A
equation of CB .
x
d. Find the slopes m2 and m3 of the other two
midsegments. Use these slopes to find
equations of the lines containing the other two sides of ¤ABC.
e. Rewrite your equations from parts (c) and (d) in slope-intercept form.
f. Use substitution to solve systems of equations to find the intersection of
each pair of lines. Plot these points A, B, and C on your graph.
Chapter 5 Properties of Triangles
Page 7 of 7
★ Challenge
37. FINDING A PATTERN In ¤ABC, the length of
C
Æ
AB is 24. In the triangle, a succession of
midsegments are formed.
•
H
J
F
At Stage 1, draw the midsegment of ¤ABC.
Æ
Label it DE.
•
At Stage 2, draw the midsegment of ¤DEC.
Æ
Label it FG.
•
At Stage 3, draw the midsegment of ¤FGC.
Æ
Label it HJ.
G
D
E
A
B
24
Copy and complete the table showing the length of the midsegment at
each stage.
0
24
Stage n
Midsegment length
1
?
2
?
3
?
4
?
5
?
38. xy USING ALGEBRA In Exercise 37, let y represent the length of the
EXTRA CHALLENGE
midsegment at Stage n. Construct a scatter plot for the data given in the table.
Then find a function that gives the length of the midsegment at Stage n.
www.mcdougallittell.com
MIXED REVIEW
SOLVING EQUATIONS Solve the equation and state a reason
for each step. (Review 2.4)
39. x º 3 = 11
40. 3x + 13 = 46
41. 8x º 1 = 2x + 17
42. 5x + 12 = 9x º 4
43. 2(4x º 1) = 14
44. 9(3x + 10) = 27
45. º2(x + 1) + 3 = 23
46. 3x + 2(x + 5) = 40
xy USING ALGEBRA Find the value of x. (Review 4.1 for 5.5)
47.
48.
(x ⴙ 2)ⴗ
(10x ⴙ 22)ⴗ
49.
4x ⴗ
xⴗ
(7x ⴙ 1)ⴗ
132ⴗ
Æ
˘ Æ
˘
38ⴗ
61ⴗ
(7x ⴙ 7)ⴗ
Æ
˘
ANGLE BISECTORS AD , BD , and CD are angle bisectors of ¤ABC.
(Review 5.2)
50. Explain why ™CAD £ ™BAD and
B
™BCD £ ™ACD.
51. Is point D the circumcenter or incenter
of ¤ABC?
Æ
Æ
G
F
D
C
Æ
52. Explain why DE £ DG £ DF.
53. Suppose CD = 10 and EC = 8. Find DF.
A
E
5.4 Midsegment Theorem
293
Page 1 of 7
5.5
Inequalities in One Triangle
What you should learn
GOAL 1 Use triangle
measurements to decide
which side is longest or
which angle is largest, as
applied in Example 2.
GOAL 2
GOAL 1
COMPARING MEASUREMENTS OF A TRIANGLE
In Activity 5.5, you may have discovered a relationship between the positions of
the longest and shortest sides of a triangle and the positions of its angles.
largest angle
shortest
side
Use the Triangle
Inequality.
Why you should learn it
RE
FE
䉲 To solve real-life
problems, such as describing
the motion of a crane as it
clears the sediment from the
mouth of a river
in Exs. 29–31.
AL LI
smallest angle
longest side
The diagrams illustrate the results stated in the theorems below.
THEOREMS
THEOREM 5.10
B
If one side of a triangle is longer than
another side, then the angle opposite
the longer side is larger than the angle
opposite the shorter side.
5
3
C
A
m™A > m™C
THEOREM 5.11
D
60ⴗ
If one angle of a triangle is larger than
another angle, then the side opposite
the larger angle is longer than the side
opposite the smaller angle.
40ⴗ
E
F
EF > DF
You can write the measurements of a triangle in order from least to greatest.
EXAMPLE 1
Writing Measurements in Order from Least to Greatest
Write the measurements of the triangles in order from least to greatest.
J
a.
b.
100ⴗ
H
45ⴗ
8
q
R
7
5
35ⴗ
G
P
SOLUTION
a. m™G < m™H < m™J
JH < JG < GH
b. QP < PR < QR
m™R < m™Q < m™P
5.5 Inequalities in One Triangle
295
Page 2 of 7
Theorem 5.11 will be proved in Lesson 5.6, using a technique called indirect
proof. Theorem 5.10 can be proved using the diagram shown below.
Proof
GIVEN 䉴 AC > AB
A
PROVE 䉴 m™ABC > m™C
Paragraph Proof Use the Ruler Postulate to
Æ
2
B
1
D
3
C
locate a point D on AC such that DA = BA.
Æ
Then draw the segment BD. In the isosceles
triangle ¤ABD, ™1 £ ™2. Because m™ABC = m™1 + m™3, it follows that
m™ABC > m™1. Substituting m™2 for m™1 produces m™ABC > m™2.
Because m™2 = m™3 + m™C, m™2 > m™C. Finally, because m™ABC > m™2
and m™2 > m™C, you can conclude that m™ABC > m™C.
..........
The proof of Theorem 5.10 above uses the fact that ™2 is an exterior angle for
¤BDC, so its measure is the sum of the measures of the two nonadjacent
interior angles. Then m™2 must be greater than the measure of either
nonadjacent interior angle. This result is stated below as Theorem 5.12.
THEOREM
THEOREM 5.12
Exterior Angle Inequality
The measure of an exterior angle of a
triangle is greater than the measure of
either of the two nonadjacent interior
angles.
A
1
C
B
m™1 > m™A and m™1 > m™B
You can use Theorem 5.10 to determine possible angle measures in a chair or
other real-life object.
Using Theorem 5.10
EXAMPLE 2
FE
L
AL I
RE
In the director’s chair shown,
AB £ AC and BC > AB. What can you conclude
about the angles in ¤ABC?
DIRECTOR’S CHAIR
Æ
Æ
SOLUTION
Æ
Æ
Because AB £ AC, ¤ABC is isosceles, so
™B £ ™C. Therefore, m™B = m™C. Because
BC > AB, m™A > m™C by Theorem 5.10. By
substitution, m™A > m™B. In addition, you can
conclude that m™A > 60°, m™B < 60°, and
m™C < 60°.
A
B
296
Chapter 5 Properties of Triangles
C
Page 3 of 7
GOAL 2
USING THE TRIANGLE INEQUALITY
Not every group of three segments can be used to form a triangle. The lengths of
the segments must fit a certain relationship.
EXAMPLE 3
Constructing a Triangle
Construct a triangle with the given group of side lengths, if possible.
a. 2 cm, 2 cm, 5 cm
b. 3 cm, 2 cm, 5 cm
c. 4 cm, 2 cm, 5 cm
SOLUTION
Try drawing triangles with the given side lengths. Only group (c) is possible. The
sum of the first and second lengths must be greater than the third length.
a.
b.
2
2
c.
3
4
2
5
2
5
5
..........
The result of Example 3 is summarized as Theorem 5.13. Exercise 34 asks you to
write a proof of this theorem.
THEOREM
THEOREM 5.13
Triangle Inequality
A
The sum of the lengths of any two sides of a triangle
is greater than the length of the third side.
AB + BC > AC
AC + BC > AB
AB + AC > BC
C
B
THEOREM
EXAMPLE 4
Finding Possible Side Lengths
A triangle has one side of 10 centimeters and another of 14 centimeters. Describe
the possible lengths of the third side.
SOLUTION
Let x represent the length of the third side. Using the Triangle Inequality, you can
write and solve inequalities.
x + 10 > 14
STUDENT HELP
Skills Review
For help with solving
inequalities, see p. 791.
x>4
䉴
10 + 14 > x
24 > x
So, the length of the third side must be greater than 4 centimeters and less than
24 centimeters.
5.5 Inequalities in One Triangle
297
Page 4 of 7
GUIDED PRACTICE
Vocabulary Check
Concept Check
Skill Check
✓
✓
✓
7
1
1. ¤ABC has side lengths of 1 inch, 1ᎏᎏ inches, and 2ᎏᎏ inches and
8
8
angle measures of 90°, 28°, and 62°. Which side is opposite each angle?
2. Is it possible to draw a triangle with side lengths of 5 inches, 2 inches, and
8 inches? Explain why or why not.
E
In Exercises 3 and 4, use the figure shown at the right.
3. Name the smallest and largest angles of ¤DEF.
18
4. Name the shortest and longest sides of ¤DEF.
GEOGRAPHY Suppose you know
the following information about distances
between cities in the Philippine Islands:
24
F
Masbate
Masbate
Cadiz to Masbate: 99 miles
Samar
PI
ES
IP
Cadiz to Guiuan: 165 miles
N
Visayan
Sea
IL
Describe the range of possible distances
from Guiuan to Masbate.
Guiuan
H
5.
103ⴗ
32ⴗ
D
P
Cadiz
Negros
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 812.
COMPARING SIDE LENGTHS Name the shortest and longest sides of the
triangle.
6.
7.
A
S
71ⴗ
C
42ⴗ
8. K
R
35ⴗ
H
50ⴗ
65ⴗ
B
J
T
COMPARING ANGLE MEASURES Name the smallest and largest angles of
the triangle.
9. A
B
15
P
10.
18
6
10
C
HOMEWORK HELP
Example 1:
Example 2:
Example 3:
Example 4:
298
Exs. 6–19
Exs. 6–19
Exs. 20–23
Exs. 24, 25
4
2
6
H
3
F
R
STUDENT HELP
11. G
8
q
xy USING ALGEBRA Use the diagram of ¤RST with exterior angle ™QRT.
T
12. Write an equation about the angle
yⴗ
measures labeled in the diagram.
13. Write two inequalities about the angle
measures labeled in the diagram.
Chapter 5 Properties of Triangles
q
xⴗ
R
zⴗ
S
Page 5 of 7
ORDERING SIDES List the sides in order from shortest to longest.
B
14.
15. E
80ⴗ
G
16.
F
30ⴗ
35ⴗ
60ⴗ
40ⴗ
A
D
C
120ⴗ
J
H
ORDERING ANGLES List the angles in order from smallest to largest.
17. L
18.
10
14
8
19. T
P
18
M
N
12
q
24
9
6
K
R
5
S
FORMING TRIANGLES In Exercises 20–23, you are given an 18 inch piece of
wire. You want to bend the wire to form a triangle so that the length of
each side is a whole number.
20. Sketch four possible isosceles triangles and label each side length.
21. Sketch a possible acute scalene triangle.
22. Sketch a possible obtuse scalene triangle.
23. List three combinations of segment lengths that will not produce triangles.
xy USING ALGEBRA In Exercises 24 and 25, solve the inequality
AB + AC > BC.
24.
25.
A
xⴙ2
B
3x ⴚ 1
xⴙ2
xⴙ3
C
xⴙ4
A
26.
C
TAKING A SHORTCUT Look at the diagram
shown. Suppose you are walking south on the
sidewalk of Pine Street. When you reach Pleasant
Street, you cut across the empty lot to go to the
corner of Oak Hill Avenue and Union Street. Explain
why this route is shorter than staying on the sidewalks.
Pleasant St.
N
Pine St.
FOCUS ON
APPLICATIONS
3x ⴚ 2
Oak Hill Ave.
B
Union St.
KITCHEN TRIANGLE In Exercises 27 and 28, use
the following information.
RE
FE
L
AL I
WORK
TRIANGLES
For ease of movement
among appliances, the
perimeter of an ideal
kitchen work triangle
should be less than 22 ft
and more than 15 ft.
refrigerator
The term “kitchen triangle” refers to the imaginary triangle
formed by three kitchen appliances: the refrigerator, the sink,
and the range. The distances shown are measured in feet.
27. What is wrong with the labels on the kitchen triangle?
28. Can a kitchen triangle have the following side lengths:
6
30ⴗ
sink
8
80ⴗ
9 feet, 3 feet, and 5 feet? Explain why or why not.
5.5 Inequalities in One Triangle
70ⴗ
4
range
299
Page 6 of 7
B
CHANNEL DREDGING In Exercises 29–31, use
the figure shown and the given information.
The crane is used in dredging mouths of rivers to clear
out the collected debris. By adjusting the length of the
boom lines from A to B, the operator of the crane can
Æ
raise and lower the boom. Suppose the mast AC is
Æ
50 feet long and the boom BC is 100 feet long.
A
100 ft
50 ft
C
29. Is the boom raised or lowered when the
boom lines are shortened?
? feet.
30. AB must be less than 31. As the boom and shovel are raised or lowered, is ™ACB ever larger than
™BAC? Explain.
INT
STUDENT HELP
NE
ER T
32.
LOGICAL REASONING In Example 4 on page 297, only two inequalities
were needed to solve the problem. Write the third inequality. Why is that
inequality not helpful in determining the range of values of x?
33.
PROOF Prove that a perpendicular segment is the shortest line
Æ
segment from a point to a line. Prove that MJ is the shortest line
¯
˘
segment from M to JN .
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with proof.
Æ
N
¯
˘
GIVEN 䉴 MJ fi JN
PROVE 䉴 MN > MJ
M
Plan for Proof Show that m™MJN > m™MNJ,
J
so MN > MJ.
34.
DEVELOPING PROOF Complete the proof
of Theorem 5.13, the Triangle Inequality.
B
GIVEN 䉴 ¤ABC
2 3
PROVE 䉴 (1) AB + BC > AC
(2) AC + BC > AB
(3) AB + AC > BC
1
D
A
Æ
C
Plan for Proof One side, say BC, is longer than or is at least as long as each
of the other sides. Then (1) and (2) are true. The proof for (3) is as follows.
Statements
1. ¤ABC
1. Given
Æ
2. Extend AC to D such that
Æ
Æ
AB £ AD.
?
3. AD + AC = 4. ™1 £ ™2
?
5. m™DBC > 6. m™DBC > m™1
7. DC > BC
? + ? > BC
8. ?
9. AB + AC > 300
Reasons
Chapter 5 Properties of Triangles
?
2. 3. Segment Addition Postulate
?
4. 5. Protractor Postulate
?
6. ?
7. 8. Substitution property of equality
9. Substitution property of equality
Page 7 of 7
Test
Preparation
QUANTITATIVE COMPARISON In Exercises 35–37, use the diagram to
choose the statement that is true about the given quantities.
A
¡
B
¡
C
¡
D
¡
The quantity in column A is greater.
The quantity in column B is greater.
The two quantities are equal.
The relationship cannot be determined from the given information.
Column A
35.
x
y
36.
x
z
37.
★ Challenge
38.
xⴗ
Column B
m
yⴗ
n
www.mcdougallittell.com
zⴗ
nⴙ3
PROOF Use the diagram
shown to prove that a perpendicular
segment is the shortest segment from
a point to a plane.
Æ
EXTRA CHALLENGE
n
m
GIVEN 䉴 PC fi plane M
P
M
D
C
PROVE 䉴 PD > PC
MIXED REVIEW
RECOGNIZING PROOFS In Exercises 39–41, look through your textbook to
find an example of the type of proof. (Review Chapters 2–5 for 5.6)
39. two-column proof
40. paragraph proof
41. flow proof
ANGLE RELATIONSHIPS Complete each statement. (Review 3.1)
? are corresponding
42. ™5 and ? .
angles. So are ™5 and 1
3 2
4
? are vertical angles.
43. ™12 and ? are alternate interior
44. ™6 and ? .
angles. So are ™6 and ? are alternate exterior
45. ™7 and 5 6
7 8
9 10
11 12
? .
angles. So are ™7 and xy USING ALGEBRA In Exercises 46–49, you are given the coordinates of
the midpoints of the sides of a triangle. Find the coordinates of the vertices
of the triangle. (Review 5.4)
46. L(º2, 1), M(2, 3), N(3, º1)
47. L(º3, 5), M(º2, 2), N(º6, 0)
48. L(3, 6), M(9, 5), N(8, 1)
49. L(3, º2), M(0, º4), N(3, º6)
5.5 Inequalities in One Triangle
301
Page 1 of 7
5.6
Indirect Proof and
Inequalities in Two Triangles
What you should learn
GOAL 1 Read and write an
indirect proof.
GOAL 2 Use the Hinge
Theorem and its converse to
compare side lengths and
angle measures.
Why you should learn it
RE
FE
䉲 To solve real-life problems,
such as deciding which of
two planes is farther from an
airport in Example 4 and
Exs. 28 and 29.
AL LI
GOAL 1
USING INDIRECT PROOF
Up to now, all of the proofs in this textbook have used the Laws of Syllogism and
Detachment to obtain conclusions directly. In this lesson, you will study indirect
proofs. An indirect proof is a proof in which you prove that a statement is true
by first assuming that its opposite is true. If this assumption leads to an
impossibility, then you have proved that the original statement is true.
EXAMPLE 1
Using Indirect Proof
Use an indirect proof to prove that a triangle
cannot have more than one obtuse angle.
SOLUTION
B
A
GIVEN 䉴 ¤ABC
C
PROVE 䉴 ¤ABC does not have more than one obtuse angle.
Begin by assuming that ¤ABC does have more than one obtuse angle.
m™A > 90° and m™B > 90°
Assume ¤ABC has two obtuse angles.
m™A + m™B > 180°
Add the two given inequalities.
You know, however, that the sum of the measures of all three angles is 180°.
m™A + m™B + m™C = 180°
Triangle Sum Theorem
m™A + m™B = 180° º m™C
Subtraction property of equality
So, you can substitute 180° º m™C for m™A + m™B in m™A + m™B > 180°.
180° º m™C > 180°
Substitution property of equality
0° > m™C
Simplify.
The last statement is not possible; angle measures in triangles cannot be negative.
䉴
So, you can conclude that the original assumption must be false. That is,
¤ABC cannot have more than one obtuse angle.
CONCEPT
SUMMARY
302
GUIDELINES FOR WRITING AN INDIRECT PROOF
1
Identify the statement that you want to prove is true.
2
Begin by assuming the statement is false; assume its opposite is true.
3
Obtain statements that logically follow from your assumption.
4
If you obtain a contradiction, then the original statement must be true.
Chapter 5 Properties of Triangles
Page 2 of 7
GOAL 2
USING THE HINGE THEOREM
Æ
Æ
In the two triangles shown, notice that AB £ DE
Æ
Æ
and BC £ EF, but m™B is greater than m™E.
C
B
122ⴗ
D
It appears that the side opposite the 122° angle
is longer than the side opposite the 85° angle.
85ⴗ
A
F
E
This relationship is guaranteed by the Hinge
Theorem below.
Exercise 31 asks you to write a proof of Theorem 5.14. Theorem 5.15 can be
proved using Theorem 5.14 and indirect proof, as shown in Example 2.
THEOREMS
THEOREM 5.14
Hinge Theorem
V
If two sides of one triangle are congruent
to two sides of another triangle, and the
included angle of the first is larger than
the included angle of the second, then
the third side of the first is longer than
the third side of the second.
THEOREM 5.15
R
80ⴗ
W
100ⴗ
S
X
T
RT > VX
Converse of the Hinge Theorem
If two sides of one triangle are congruent
to two sides of another triangle, and the
third side of the first is longer than the
third side of the second, then the included
angle of the first is larger than the included
angle of the second.
B
E
D
A
8
7
F
C
m™A > m™D
EXAMPLE 2
Indirect Proof of Theorem 5.15
Æ
Æ
Æ
Æ
GIVEN 䉴 AB £ DE
E
BC £ EF
AC > DF
B
PROVE 䉴 m™B > m™E
D
F
A
C
SOLUTION Begin by assuming that m™B ⬎ m™E. Then, it follows that either
STUDENT HELP
Study Tip
The symbol ⬎ is read as
“is not greater than.”
m™B = m™E or m™B < m™E.
Case 1 If m™B = m™E, then ™B £ ™E. So, ¤ABC £ ¤DEF by the
SAS Congruence Postulate and AC = DF.
Case 2
If m™B < m™E, then AC < DF by the Hinge Theorem.
Both conclusions contradict the given information that AC > DF. So the original
assumption that m™B ⬎ m™E cannot be correct. Therefore, m™B > m™E.
5.6 Indirect Proof and Inequalities in Two Triangles
303
Page 3 of 7
EXAMPLE 3
Finding Possible Side Lengths and Angle Measures
You can use the Hinge Theorem and its converse to choose possible side lengths
or angle measures from a given list.
Æ
Æ Æ
Æ
a. AB £ DE, BC £ EF, AC = 12 inches, m™B = 36°, and m™E = 80°. Which
Æ
of the following is a possible length for DF: 8 in., 10 in., 12 in., or 23 in.?
Æ
ÆÆ
Æ
b. In a ¤RST and a ¤XYZ, RT £ XZ, ST £ YZ, RS = 3.7 centimeters,
XY = 4.5 centimeters, and m™Z = 75°. Which of the following is a possible
measure for ™T: 60°, 75°, 90°, or 105°?
SOLUTION
a. Because the included angle in
¤DEF is larger than the included
Æ
angle in ¤ABC, the third side DF
Æ
must be longer than AC. So, of the
four choices, the only possible
Æ
length for DF is 23 inches.
A diagram of the triangles shows
that this is plausible.
B
E
36ⴗ
80ⴗ
D
F
12 in.
A
C
b. Because the third side in ¤RST is shorter than the third side in ¤XYZ, the
included angle ™T must be smaller than ™Z. So, of the four choices, the
only possible measure for ™T is 60°.
EXAMPLE 4
FOCUS ON
CAREERS
Comparing Distances
TRAVEL DISTANCES You and a friend are flying separate planes. You leave the
airport and fly 120 miles due west. You then change direction and fly W 30° N
for 70 miles. (W 30° N indicates a north-west direction that is 30° north of due
west.) Your friend leaves the airport and flies 120 miles due east. She then
changes direction and flies E 40° S for 70 miles. Each of you has flown
190 miles, but which plane is farther from the airport?
SOLUTION
Begin by drawing a diagram, as shown below. Your flight is represented by
¤PQR and your friend’s flight is represented by ¤PST.
you
R
70 mi
RE
FE
L
AL I
INT
NE
ER T
CAREER LINK
www.mcdougallittell.com
304
airport
P
150ⴗ
œ
AIR TRAFFIC
CONTROLLERS
help ensure the safety of
airline passengers and
crews by developing air
traffic flight paths that keep
planes a safe distance apart.
N
120 mi
E
W
120 mi
S
140 ⴗ
70 mi
S
T
your friend
Because these two triangles have two sides that are congruent, you can apply the
Æ
Æ
Hinge Theorem to conclude that RP is longer than TP.
䉴
So, your plane is farther from the airport than your friend’s plane.
Chapter 5 Properties of Triangles
Page 4 of 7
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Skill Check
✓
1. Explain why an indirect proof might also be called a proof by contradiction.
2. To use an indirect proof to show that two lines m and n are parallel, you
would first make the assumption that ? .
In Exercises 3–5, complete with <, >, or =.
? m™2
3. m™1 ? NQ
4. KL ? FE
5. DC N
E
D
38ⴗ
27
26
1
47ⴗ
K
2
q
P
37ⴗ
45ⴗ
L
F
C
M
6. Suppose that in a ¤ABC, you want to prove that BC > AC. What are the two
cases you would use in an indirect proof?
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 812.
USING THE HINGE THEOREM AND ITS CONVERSE Complete with <, >, or =.
? TU
7. RS ? m™2
8. m™1 ? m™2
9. m™1 1
S
110ⴗ
U
15
1
R
13
2
130ⴗ
T
2
? ZY
10. XY ? m™2
11. m™1 Z
? m™2
12. m™1 1
1 2
W
38ⴗ
41ⴗ
2
Y
11
X
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 21–24
Example 2: Exs. 25–27
Example 3: Exs. 7–17
Example 4: Exs. 28, 29
? CB
13. AB D 20ⴗ
? SV
14. UT A
13
? m™2
15. m™1 T
S
1
44ⴗ
B 18ⴗ
C
9
E
2
45ⴗ
V
8
U
5.6 Indirect Proof and Inequalities in Two Triangles
305
Page 5 of 7
LOGICAL REASONING In Exercises 16 and 17, match the given
information with conclusion A, B, or C. Explain your reasoning.
A. AD > CD
B. AC > BD
16. AC > AB, BD = CD
C. m™4 < m™5
17. AB = DC, m™3 < m™5
A
A
3 2
3 2
C
6
B
4 5
D
1
C
6
4 5
D
1
B
xy USING ALGEBRA Use an inequality to describe a restriction on the
value of x as determined by the Hinge Theorem or its converse.
18.
x
19.
20.
4
12 45ⴗ
9
60ⴗ
70ⴗ 8
8
3x ⴙ 1
3
115ⴗ
xⴙ3
65ⴗ
12
(4x ⴚ 5)ⴗ
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with negations in
Exs. 21–23.
2
3
ASSUMING THE NEGATION OF THE CONCLUSION In Exercises 21–23, write
the first statement for an indirect proof of the situation.
21. If RS + ST ≠ 12 in. and ST = 5 in., then RS ≠ 7 in.
Æ
Æ
22. In ¤MNP, if Q is the midpoint of NP, then MQ is a median.
23. In ¤ABC, if m™A + m™B = 90°, then m™C = 90°.
24.
DEVELOPING PROOF Arrange statements A–D in correct order to write
an indirect proof of Postulate 7 from page 73: If two lines intersect, then their
intersection is exactly one point.
GIVEN 䉴 line m, line n
PROVE 䉴 Lines m and n intersect in exactly one point.
A. But this contradicts Postulate 5, which states that there is exactly one line
through any two points.
B. Then there are two lines (m and n) through points P and Q.
C. Assume that there are two points, P and Q, where m and n intersect.
D. It is false that m and n can intersect in two points, so they must intersect
in exactly one point.
25.
PROOF Write an indirect proof of Theorem 5.11 on page 295.
GIVEN 䉴 m™D > m™E
D
E
PROVE 䉴 EF > DF
Plan for Proof In Case 1, assume that EF < DF.
In Case 2, assume that EF = DF. Show that neither
case can be true, so EF > DF.
306
Chapter 5 Properties of Triangles
F
Page 6 of 7
PROOF Write an indirect proof in paragraph form. The diagrams, which
illustrate negations of the conclusions, may help you.
Æ
26. GIVEN 䉴 ™1 and ™2 are
27. GIVEN 䉴 RU is an altitude,
Æ˘
supplementary.
PROVE 䉴 m ∞ n
RU bisects ™SRT.
PROVE 䉴 ¤RST is isosceles.
t
n
R
m
1
3
2
U
S
Begin by assuming that m Ω n.
T
Begin by assuming that RS > RT.
COMPARING DISTANCES In Exercises 28 and 29, consider the flight
paths described. Explain how to use the Hinge Theorem to determine who
is farther from the airport.
28. Your flight: 100 miles due west, then 50 miles N 20° W
Friend’s flight: 100 miles due north, then 50 miles N 30° E
29. Your flight: 210 miles due south, then 80 miles S 70° W
Friend’s flight: 80 miles due north, then 210 miles N 50° E
Test
Preparation
30. MULTI-STEP PROBLEM Use the diagram of the tank cleaning system’s
expandable arm shown below.
Æ
a. As the cleaning system arm expands, ED gets longer. As ED increases,
what happens to m™EBD? What happens to m™DBA?
Æ
b. Name a distance that decreases as ED gets longer.
c.
Writing
Explain how the cleaning arm illustrates the Hinge Theorem.
B
E
★ Challenge
31.
A
D
PROOF Prove Theorem 5.14, the Hinge Theorem.
Æ
Æ Æ
Æ
GIVEN 䉴 AB £ DE , BC £ EF ,
B
C
E
F
m™ABC > m™DEF
PROVE 䉴 AC > DF
Plan for Proof
A
H
D
P
1. Locate a point P outside ¤ABC so you can construct ¤PBC £ ¤DEF.
2. Show that ¤PBC £ ¤DEF by the SAS Congruence Postulate.
Æ
Æ˘
3. Because m™ABC > m™DEF, locate a point H on AC so that BH
bisects ™PBA.
EXTRA CHALLENGE
www.mcdougallittell.com
4. Give reasons for each equality or inequality below to show that AC > DF.
AC = AH + HC = PH + HC > PC = DF
5.6 Indirect Proof and Inequalities in Two Triangles
307
Page 7 of 7
MIXED REVIEW
CLASSIFYING TRIANGLES State whether the triangle described is
isosceles, equiangular, equilateral, or scalene. (Review 4.1 for 6.1)
32. Side lengths:
33. Side lengths:
34. Side lengths:
3 cm, 5 cm, 3 cm
5 cm, 5 cm, 5 cm
5 cm, 6 cm, 8 cm
35. Angle measures:
36. Angle measures:
37. Angle measures:
60°, 60°, 60°
65°, 50°, 65°
30°, 30°, 120°
xy USING ALGEBRA In Exercises 38–41, use the diagram
shown at the right. (Review 4.1 for 6.1)
D
38. Find the value of x.
39. Find m™B.
40. Find m™C.
41. Find m™BAC.
A
3xⴗ
(x ⴙ 19)ⴗ
42. DESCRIBING A SEGMENT Draw any equilateral
B
triangle ¤RST. Draw a line segment from vertex R to
Æ
the midpoint of side ST . State everything that you know
about the line segment you have drawn. (Review 5.3)
QUIZ 2
C
(x ⴙ 13)ⴗ
Self-Test for Lessons 5.4– 5.6
In Exercises 1–3, use the triangle shown at the
right. The midpoints of the sides of ¤CDE are F, G,
and H. (Lesson 5.4)
E
H
G
Æ
1. FG ∞ ?
C
F
D
2. If FG = 8, then CE = ? .
3. If the perimeter of ¤CDE = 42, then the perimeter of ¤GHF = ? .
In Exercises 4–6, list the sides in order from shortest to longest. (Lesson 5.5)
4.
5.
L
75ⴗ
6. M
M
75ⴗ
74ⴗ
q
q
50ⴗ
7. In ¤ABC and ¤DEF shown at the right,
Æ
Æ
which is longer, AB or DE? (Lesson 5.6)
49ⴗ
Hikers in the Grand Canyon
308
P
P
C
B
72ⴗ
A
8.
48ⴗ
M
E
D
73ⴗ
F
HIKING Two groups of hikers leave from the same base camp and head
in opposite directions. The first group walks 4.5 miles due east, then changes
direction and walks E 45° N for 3 miles. The second group walks 4.5 miles
due west, then changes direction and walks W 20° S for 3 miles. Each group
has walked 7.5 miles, but which is farther from the base camp? (Lesson 5.6)
Chapter 5 Properties of Triangles
N
Page 1 of 5
CHAPTER
5
Chapter Summary
WHAT did you learn?
WHY did you learn it?
Use properties of perpendicular bisectors and
angle bisectors. (5.1)
Decide where a hockey goalie should be positioned
to defend the goal. (p. 270)
Use properties of perpendicular bisectors and
angle bisectors of a triangle. (5.2)
Find the center of a mushroom ring. (p. 277)
Use properties of medians and altitudes of a
triangle. (5.3)
Find points in a triangle used to measure a
person’s heart fitness. (p. 283)
Use properties of midsegments of a triangle. (5.4)
Determine the length of the crossbar of a swing set.
(p. 292)
Compare the lengths of the sides or the measures
of the angles of a triangle. (5.5)
Determine how the lengths of the boom lines of a
crane affect the position of the boom. (p. 300)
Understand and write indirect proofs. (5.6)
Prove theorems that cannot be easily proved
directly.
Use the Hinge Theorem and its converse to
compare side lengths and angle measures of
triangles. (5.6)
Decide which of two airplanes is farther from an
airport. (p. 304)
How does Chapter 5 fit into the BIGGER PICTURE of geometry?
In this chapter, you studied properties of special segments of triangles,
which are an important building block for more complex figures that you
will explore in later chapters. The special segments of a triangle have
applications in many areas such as demographics (p. 280), medicine
(p. 283), and room design (p. 299).
STUDY STRATEGY
Did you test your
memory?
The list of important vocabulary
terms and skills you made,
following the Study Strategy
on page 262, may resemble
this one.
Memory Test
perpendicular bisector
XM = YM
Æ
k fi XY
k
X
M
Y
perpendicular bisector
of a triangle
angle bisector of
a triangle
309
Page 2 of 5
Chapter Review
CHAPTER
5
• perpendicular bisector,
• equidistant from two lines,
p. 264
• equidistant from two points,
p. 264
p. 273
• perpendicular bisector of a
• angle bisector of a triangle,
triangle, p. 272
• distance from a point to a
p. 274
• concurrent lines, p. 272
• point of concurrency, p. 272
line, p. 266
5.1
• circumcenter of a triangle,
p. 266
• centroid of a triangle, p. 279
• altitude of a triangle, p. 281
• orthocenter of a triangle,
p. 281
• incenter of a triangle, p. 274
• median of a triangle, p. 279
• midsegment of a triangle,
p. 287
indirect
proof, p. 302
•
Examples on
pp. 264–267
PERPENDICULARS AND BISECTORS
Æ˘
In the figure, AD is the angle bisector of
Æ
™BAC and the perpendicular bisector of BC. You know that
BE = CE by the definition of perpendicular bisector and that
AB = AC by the Perpendicular Bisector Theorem. Because
Æ˘
Æ˘
Æ
Æ
DP fi AP and DQ fi AQ , then DP and DQ are the distances
from D to the sides of ™PAQ and you know that DP = DQ
by the Angle Bisector Theorem.
EXAMPLES
D
q
E
P
B
C
A
In Exercises 1–3, use the diagram.
Æ
˘
Æ
1. If SQ is the perpendicular bisector of RT, explain how you know that
Æ
Æ
Æ
R
Æ
RQ £ TQ and RS £ TS .
Æ
Æ
U
2. If UR £ UT, what can you conclude about U?
Æ˘
Æ˘
3. If Q is equidistant from SR and ST , what can you conclude about Q?
5.2
q
S
T
Examples on
pp. 272–274
BISECTORS OF A TRIANGLE
EXAMPLES The perpendicular bisectors of a triangle intersect at the
circumcenter, which is equidistant from the vertices of the triangle. The
angle bisectors of a triangle intersect at the incenter, which is equidistant
from the sides of the triangle.
4. The perpendicular bisectors of ¤RST
intersect at K. Find KR.
R
K
12
5. The angle bisectors of ¤XYZ intersect at W.
Find WB.
S
Z
32
A
8
T
310
Chapter 5 Properties of Triangles
Y
B
W
10
X
Page 3 of 5
5.3
Examples on
pp. 279–281
MEDIANS AND ALTITUDES OF A TRIANGLE
EXAMPLES The medians of a triangle intersect at the centroid. The lines
containing the altitudes of a triangle intersect at the orthocenter.
¯˘ ¯˘
¯˘
HN , JM , and KL intersect at Q.
B
2
3
AP = ᎏᎏAD
F
H
N
J
q
D
P
M
L
A
E
K
C
Name the special segments and point of concurrency of the triangle.
6.
7.
7
8.
6
7
9.
6
8
8
¤XYZ has vertices X(0, 0), Y(º4, 0), and Z(0, 6). Find the coordinates of the
indicated point.
10. the centroid of ¤XYZ
5.4
11. the orthocenter of ¤XYZ
Examples on
pp. 287–289
MIDSEGMENT THEOREM
A midsegment of a triangle connects
the midpoints of two sides of the triangle. By the
Midsegment Theorem, a midsegment of a triangle
is parallel to the third side and its length is half the
length of the third side.
EXAMPLES
Æ
Æ
1
2
DE ∞ AB , DE = ᎏᎏAB
C
E
D
5
B
10
A
In Exercises 12 and 13, the midpoints of the sides of ¤HJK are L(4, 3),
M(8, 3), and N(6, 1).
12. Find the coordinates of the vertices of the triangle.
13. Show that each midsegment is parallel to a side of the triangle.
14. Find the perimeter of ¤BCD.
15. Find the perimeter of ¤STU.
B
T
G
E
12
D
R
F
22
9
C
U
10
P
S
24
9
q
Chapter Review
311
Page 4 of 5
5.5
Examples on
pp. 295–297
INEQUALITIES IN ONE TRIANGLE
EXAMPLES In a triangle, the side and the angle of greatest
measurement are always opposite each other. In the diagram,
Æ
the largest angle, ™MNQ, is opposite the longest side, MQ.
M
41.4ⴗ
5
By the Exterior Angle Inequality,
m™MQP > m™N and m™MQP > m™M.
6
55.8ⴗ 124.2ⴗ
82.8ⴗ
By the Triangle Inequality, MN + NQ > MQ,
NQ + MQ > MN, and MN + MQ > NQ.
q
4
N
P
In Exercises 16–19, write the angle and side measurements in order from least
to greatest.
16.
17.
C
25
10
20.
50ⴗ
19.
H
K
23
55ⴗ
9
70ⴗ
35
D
A
18. J
F
E
G
L
M
B
8
FENCING A GARDEN You are enclosing a triangular garden region with a
fence. You have measured two sides of the garden to be 100 feet and 200 feet.
What is the maximum length of fencing you need? Explain.
5.6
Examples on
pp. 302–304
INDIRECT PROOF AND INEQUALITIES IN TWO TRIANGLES
EXAMPLES
Æ
Æ
Æ
Æ
AB £ DE and BC £ EF
E
F
Hinge Theorem: If m™E > m™B,
then DF > AC.
B
C
D
Converse of the Hinge Theorem: If DF > AC,
then m™E > m™B.
A
In Exercises 21–23, complete with <, >, or =.
? CB
21. AB ? m™2
22. m™1 C
R
? VS
23. TU 16
S
126ⴗ
D 92ⴗ
88ⴗ
A
U
T
1
2
B
P
15
q
S
24. Write the first statement for an indirect proof of this situation: In a ¤MPQ, if
™M £ ™Q, then ¤MPQ is isosceles.
25. Write an indirect proof to show that no triangle has two right angles.
312
W
126ⴗ
Chapter 5 Properties of Triangles
V
Page 5 of 5
CHAPTER
5
Chapter Test
In Exercises 1–5, complete the statement with the word always,
sometimes, or never.
1. If P is the circumcenter of ¤RST, then PR, PS, and PT are ? equal.
Æ˘
Æ
Æ
2. If BD bisects ™ABC, then AD and CD are ? congruent.
3. The incenter of a triangle ? lies outside the triangle.
4. The length of a median of a triangle is ? equal to the length of a midsegment.
Æ
Æ
Æ
Æ
5. If AM is the altitude to side BC of ¤ABC, then AM is ? shorter than AB.
In Exercises 6–10, use the diagram.
C
6. Find each length.
a. HC
b. HB
c. HE
d. BC
E
F
7. Point H is the ? of the triangle.
9.9
H
Æ
6
8. CG is a(n) ? , ? , ? , and ? of ¤ABC .
A
Æ
9. EF = ? and EF ∞ ? by the ? Theorem.
G
8
B
10. Compare the measures of ™ACB and ™BAC. Justify your answer.
11.
LANDSCAPE DESIGN You are designing a circular
swimming pool for a triangular lawn surrounded by
apartment buildings. You want the center of the pool to be
equidistant from the three sidewalks. Explain how you can
locate the center of the pool.
In Exercises 12–14, use the photo of the three-legged tripod.
12. As the legs of a tripod are spread apart, which theorem
guarantees that the angles between each pair of legs get larger?
13. Each leg of a tripod can extend to a length of 5 feet. What is
the maximum possible distance between the ends of two legs?
Æ Æ
Æ
14. Let OA, OB, and OC represent the legs of a tripod. Draw
and label a sketch. Suppose the legs are congruent and
Æ
Æ
m™AOC > m™BOC. Compare the lengths of AC and BC.
In Exercises 15 and 16, use the diagram at the right.
15. Write a two-column proof.
B
A
GIVEN 䉴 AC = BC
PROVE 䉴 BE < AE
C
16. Write an indirect proof.
GIVEN 䉴 AD ≠ AB
PROVE 䉴 m™D ≠ m™ABC
E
D
Chapter Test
313