Perpendicular
and Angle
Bisectors
Bisectors & Points
of Concurrency
Unit 1 Day 26
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
GeometryGeometry
Holt
Perpendicular and Angle Bisectors
Warm Up
Construct each of the following.
1. A perpendicular bisector.
2. An angle bisector.
3. Find the midpoint and slope of the segment
(2, 8) and (–4, 6).
Holt McDougal Geometry
Perpendicular and Angle Bisectors
When a point is the same distance from two or more
objects, the point is said to be equidistant from
the objects.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 1A: Applying the Perpendicular Bisector
Theorem and Its Converse
Find each measure.
MN
MN = LN
 Bisector Thm.
MN = 2.6
Substitute 2.6 for LN.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 1B: Applying the Perpendicular Bisector
Theorem and Its Converse
Find each measure.
BC
Since AB = AC and
, is the
perpendicular bisector of
by
the Converse of the Perpendicular
Bisector Theorem.
BC = 2CD
Def. of seg. bisector.
BC = 2(12) = 24
Substitute 12 for CD.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 1C: Applying the Perpendicular Bisector
Theorem and Its Converse
Find each measure.
TU
TU = UV
 Bisector Thm.
3x + 9 = 7x – 17
Substitute the given values.
9 = 4x – 17
Subtract 3x from both sides.
26 = 4x
6.5 = x
Add 17 to both sides.
Divide both sides by 4.
So TU = 3(6.5) + 9 = 28.5.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Remember that the distance between a point and a
line is the length of the perpendicular segment from
the point to the line.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 2A: Applying the Angle Bisector Theorem
Find the measure.
BC
BC = DC
 Bisector Thm.
BC = 7.2
Substitute 7.2 for DC.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 2B: Applying the Angle Bisector Theorem
Find the measure.
mEFH, given that mEFG = 50°.
Since EH = GH,
and
,
bisects
EFG by the Converse
of the Angle Bisector Theorem.
Def. of  bisector
Substitute 50° for mEFG.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 2C: Applying the Angle Bisector Theorem
Find mMKL.
Since, JM = LM,
and
,
bisects JKL
by the Converse of the Angle
Bisector Theorem.
mMKL = mJKM
Def. of  bisector
3a + 20 = 2a + 26
a + 20 = 26
a=6
Substitute the given values.
Subtract 2a from both sides.
Subtract 20 from both sides.
So mMKL = [2(6) + 26]° = 38°
Holt McDougal Geometry
Perpendicular and Angle Bisectors
When three or more lines intersect at one point, the
lines are said to be concurrent. The point of
concurrency is the point where they intersect.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
A circle that contains all the vertices of a polygon is
circumscribed about the polygon.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
A circle inscribed in a polygon intersects each line
that contains a side of the polygon at exactly one
point.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
A median of a triangle is a segment whose
endpoints are a vertex of the triangle and the
midpoint of the opposite side.
Every triangle has three medians, and the medians
are concurrent.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
The point of concurrency of the medians of a triangle
is the centroid of the triangle . The centroid is
always inside the triangle. The centroid is also called
the center of gravity because it is the point where a
triangular region will balance.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 1A: Using the Centroid to Find Segment
Lengths
In ∆LMN, RL = 21 and SQ =4.
Find LS.
Centroid Thm.
Substitute 21 for RL.
LS = 14
Holt McDougal Geometry
Simplify.
Perpendicular and Angle Bisectors
Example 1B: Using the Centroid to Find Segment
Lengths
In ∆LMN, RL = 21 and SQ =4.
Find NQ.
Centroid Thm.
NS + SQ = NQ
Seg. Add. Post.
Substitute
Subtract
NQ for NS.
from both sides.
Substitute 4 for SQ.
12 = NQ
Holt McDougal Geometry
Multiply both sides by 3.
Perpendicular and Angle Bisectors
Check It Out! Example 1a
In ∆JKL, ZW = 7, and LX = 8.1.
Find KW.
Centroid Thm.
Substitute 7 for ZW.
KW = 21
Holt McDougal Geometry
Multiply both sides by 3.
Perpendicular and Angle Bisectors
Check It Out! Example 1b
In ∆JKL, ZW = 7, and LX = 8.1.
Find LZ.
Centroid Thm.
Substitute 8.1 for LX.
LZ = 5.4
Holt McDougal Geometry
Simplify.
Perpendicular and Angle Bisectors
Example 2: Problem-Solving Application
A sculptor is shaping a
triangular piece of iron that
will balance on the point of a
cone. At what coordinates will
the triangular region balance?
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 2 Continued
1
Understand the Problem
The answer will be the coordinates of the
centroid of the triangle. The important
information is the location of the vertices,
A(6, 6), B(10, 7), and C(8, 2).
2
Make a Plan
The centroid of the triangle is the point of
intersection of the three medians. So write the
equations for two medians and find their point of
intersection.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 2 Continued
3
Solve
Let M be the midpoint of AB and N be the
midpoint of AC.
CM is vertical. Its equation is x = 8. BN has slope 1.
Its equation is y = x – 3. The coordinates of the
centroid are D(8, 5).
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Example 2 Continued
4
Look Back
Let L be the midpoint of BC. The equation for AL
is
Holt McDougal Geometry
, which intersects x = 8 at D(8, 5).
Perpendicular and Angle Bisectors
Check It Out! Example 2
Find the average of the x-coordinates and the
average of the y-coordinates of the vertices of
∆PQR. Make a conjecture about the centroid of a
triangle.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Check It Out! Example 2 Continued
The x-coordinates are 0, 6 and 3. The average
is 3.
The y-coordinates are 8, 4 and 0. The average
is 4.
The x-coordinate of the centroid is the average
of the x-coordinates of the vertices of the ∆,
and the y-coordinate of the centroid is the
average of the y-coordinates of the vertices of
the ∆.
Holt McDougal Geometry
Perpendicular and Angle Bisectors
Assignment
Page 180 #2-7
Page 193-195 #1, 3-7, 21-26
Holt McDougal Geometry