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Chapter 11 Algebra and Graphing
11.6 Coordinate Geometry and Networks
NCTM Standards
• use coordinate geometry to represent and examine the properties of geometric
shapes (6–8)
• use coordinate geometry to examine special geometric shapes, such as regular
polygons or those with pairs of parallel or perpendicular sides (6–8)
• use visual tools such as networks to represent and solve problems (6–8)
Coordinate geometry establishes a connection between algebra and geometry. This
enables mathematicians to employ algebraic methods to solve geometry problems and to
use geometric models to gain insights into algebra problems.
The Distance Formula
How can we find the length of a line segment using coordinates?
LE 1 Connection
(a) Plot D(2, 1), E(5, 3), and F(5, 1) on a graph, and draw right triangle DEF.
(b) The legs of DEF have lengths _______ and _______.
(c) Find DE using the Pythagorean Theorem.
LE 2 Skill
A seventh grader computes DE in LE 1 as 32 22 3 2 5. Is this right? If
not, what would you tell the child?
The Pythagorean Theorem is the basis for computing the distance between any two
points on a two-dimensional graph. Complete the following exercise to derive the distance formula.
H
B(x2 , y2)
A(x1, y1)
C
LE 3 Reasoning
(a) For two points A(x1, y1) and B(x2, y2), construct a right triangle with hypotenuse
AB as shown in Figure 11–21.
(b) What are the coordinates of C?
(c) Give the lengths AC and BC in terms of the coordinates.
(d) Using the results of part (c), (AB)2 (AC)2 (BC)2 _______.
(e) Solve for AB, and write it in terms of the coordinates.
(f) Does part (e) involve induction or deduction?
Figure 11–21
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11.6 Coordinate Geometry and Networks
9
If the slope of AB is negative, the derivation is similar and the resulting formula is
the same. The distance formula tells us how to find the length of a line segment using
the coordinates of its endpoints.
The Distance Formula
The distance between two points A(x1, y1) and B(x2, y2) is
AB (x2 x1)2 (y2 y1)2
LE 4 Connection
A quadrilateral ABCD has coordinates A(0, 0), B(5, 0), C(8, 4), and D(3, 4). Use the
distance formula to show that ABCD is a rhombus.
The Midpoint Formula
How are the coordinates of the midpoint of a line segment related to the coordinates of the
endpoints?
H
LE 5 Reasoning
Plot the points A(3, 2), B(1, 4), and C(5, 6), and draw ABC on graph paper.
Find the coordinates of the midpoint of AC.
Find the coordinates of the midpoint of AB.
Find the coordinates of the midpoint of BC.
What is the relationship between the coordinates of the endpoints of a line
segment and the coordinates of the midpoint?
(f) Given two points A(x1, y1) and B(x2, y2), what are the coordinates of the
midpoint of AB?
(g) Using parts (b)–(d) to answer parts (e) and (f) is an example of what kind of
reasoning?
(a)
(b)
(c)
(d)
(e)
The answer to LE 5(f) is the midpoint formula.
The Midpoint Formula
Given A(x1, y1) and B(x2, y2), then the midpoint M of AB is
x 2 x , y 2 y 1
2
1
2
The midpoint and distance formulas can be used to find properties of triangles and
quadrilaterals.
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Chapter 11 Algebra and Graphing
LE 6 Reasoning
H
ABC is a right triangle with A(0, 0), B(8, 0), and C(0, 6).
(a)
(b)
(c)
(d)
(e)
y
C (a + b, c)
D (b, c)
A (0, 0)
In Chapter 8, you studied properties of parallelograms. The following exercise
shows how to use coordinate geometry to prove one such property: that the diagonals of
a parallelogram bisect each other.
x
B (a, 0)
Plot ABC on a coordinate graph.
Find the coordinates of M, the midpoint of the hypotenuse.
Compare the lengths AM, BM, and CM.
Repeat parts (a)–(c) with a new right triangle.
Make a generalization based on your results.
Figure 11–22
LE 7 Reasoning
H
(a) What is the definition of a parallelogram?
(b) Prove that ABCD in Figure 11–22 is a parallelogram.
(c) Prove that the midpoint of AC is the midpoint of BD.
Networks
Suppose you live in Chicago, and you want to take a trip that includes Detroit, St. Louis,
and Pittsburgh, in any order, and then return home. What is the fastest route? You can
solve this problem by drawing a graph of the four cities (see Figure 11–23) called a network. A network is a graph whose edges are labeled with numbers such as lengths or
times.
Chicago 3.5 Detroit
5.5 hr
10
10
6 hr
12 hr
St. Louis
Pittsburgh
Figure 11–23
Allview
5
2.
Cool City
4
3.
2.8
3 Boomtown
2.7
2.1
Dullsville
Figure 11–24
H
LE 8 Reasoning
H
LE 9 Reasoning
Find the fastest route that starts in Chicago and goes through all 3 cities and then back
to Chicago.
A phone company wants to link up 4 cities (Figure 11–24). The network shows the
cost (in millions of dollars) of different possible links. Find the cheapest possible
network.
The points in a network are called vertices, and the curves are called edges. A
network is traversable if it is possible to pass through each edge exactly one time.
You do not have to end up at the vertex where you started. What makes a network
traversable?
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11.6 Coordinate Geometry and Networks
H
11
LE 10 Reasoning
(a) Which of the following networks are traversable? Mark where you started and
where you finished.
(1)
(2)
(4)
(5)
(3)
(b) Determine whether the vertices of each network are odd or even. An even number
of arcs meet at an even vertex, and an odd number of arcs meet at an odd vertex.
(c) Based on part (b), make a conjecture about a property of vertices that makes a
network traversable.
To determine if a network is traversable, check whether each vertex is even or odd.
Traversable Network
A network is traversable if
1. it has all even vertices or
2. it has exactly two odd vertices. One of these will be where you start, and the
other will be where you finish.
A business sometimes desires a traversable network where one can start and end in
the same place.
H
LE 11 Reasoning
A street cleaner wants to find a route that travels along each street exactly once and
starts and ends at the same place. Such a route is called an Euler circuit. What property of the vertices makes a network an Euler circuit?
Since an Euler circuit is traversable and has the same starting and ending point, it
must have all even vertices.
H
LE 12 Reasoning
(a) Tell which of the following has an Euler circuit from point A.
A
Route 1
A
Route 2
(b) Would it be possible to make an Euler circuit from a different starting point in
either network?
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Chapter 11 Algebra and Graphing
H
LE 13 Reasoning
(1)
(2)
(3)
(a) Which of the networks are traversable?
(b) Which of the networks are Euler circuits?
LE 14 Summary
(a) Tell what you learned about coordinate geometry in this section. How is the
distance formula related to the Pythagorean Theorem?
(b) Tell what you learned about networks. What are traversable networks and Euler
circuits?
Answers to Selected Lesson Exercises
(c) 13
1. (b) 2; 3
2. No. Do it the correct way, and compare the answers.
3. (b) (x2, y1)
(c) x2 x1 and y2 y1
(d) (x2 x1)2 (y2 y1)2
(e) (x2 x1)2 (y2 y1)2
(f) Deduction
4. AB BC CD DA 5, so ABCD is a rhombus.
5. (b) (4, 4)
(c) (1, 3)
(d) (2, 5)
(e) The coordinates of the midpoint are the averages
of the coordinates of the two endpoints.
(f)
x 2 x , y 2 y 1
2
1
2
(g) Inductive
6. (b) M(4, 3)
(c) AM BM CM 5
(e) The midpoint of the hypotenuse is equidistant
from the three vertices.
7. (b) Hint: Compute slopes to show that the opposite
sides are parallel.
ab c
, .
(c) The midpoints of both diagonals are
2 2
8. 27 hours; possible route: Chicago, Detroit, Pittsburgh,
St. Louis, Chicago
9. $7.4 million; A to C, B to C, D to C
10. (a) (1), (2), (4)
(b) (1) has 3 even vertices.
(2) has 1 even vertex and 2 odd vertices.
(3) has 4 odd vertices.
(4) has 2 even and 2 odd vertices.
(5) has 4 odd vertices.
(c) Answer follows the exercise.
11. Answer follows the exercise.
12. (a) Route 1
(b) Yes; you could start at any point on route 1.
13. (a) (1), (2)
(b) (2)
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11.6 Coordinate Geometry and Networks
13
11.6 Homework Exercises
Basic Exercises
H
H 10.
y
1. Explain why the distance between A(x1, y1) and
B(x2, y2) is
D (0, a)
C
AB (x2 x1)2 (y2 y1)2
y
A
B (x2, y2)
ABCD is a square.
(a) What must the coordinates of C be?
(b) What property do the diagonals have? Prove it.
(c) Name a second property that the diagonals have.
Prove it.
(d) Name a third property that the diagonals have.
Prove it.
A (x1, y1)
x
2. Explain how the distance formula is related to the
Pythagorean Theorem.
3. Find the perimeter of ABC with coordinates
A(3, 6), B(3, 9), and C(5, 7).
H
H
8. Point M is the midpoint of AB. Point A has coordinates (3, 7), and M has coordinates (2, 1). Find the
coordinates of B.
9. ABCD is a rectangle.
y
D (0, b)
A (0, 0)
C
B (a, 0) x
(a) What are the coordinates of C?
(b) What property do the diagonals have? Prove it.
(c) Name a second property that the diagonals have.
Prove it.
y
B (a, b)
A
C (c, 0) x
5. Three vertices of a parallelogram are (1, 2), (0, 2),
and (2, 0). Find three possible coordinates for the
fourth vertex.
7. Point A has coordinates (4, 1), and point B has
coordinates (2, 7). What are the coordinates of the
mid-point of AB?
H
H 11.
4. Determine whether or not (0, 4), (4, 5), and (6, 3)
are the vertices of a right triangle.
6. Three vertices of a rhombus are (1, 2), (4, 1), and
(3, 2). What are the coordinates of the fourth vertex?
B (a, 0) x
(a) Find the coordinates of the midpoints D of AB
and E of BC.
(b) Show that DE ´ AC.
(c) How does DE compare to AC? Prove it.
H 12.
y
D(0, b) G
C
H
A
F
B (a, 0) x
E
(a) Find the coordinates of the midpoints of the four
sides, E, F, G, and H.
(b) What kind of shape is EFGH? Prove it.
13. Lyle is at home. He wants to go to the park, store,
and library in any order and then return home. What
is the fastest route he could take? How many different routes of the minimum time are there?
Home
10
5 min.
10
5
Park
10 min.
12
Store
Library
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Chapter 11 Algebra and Graphing
14. A phone company wants to link up 5 cities. The
network shows the cost (in millions of dollars) of
different possible links. Find the cheapest possible
network.
Quagmire
Pipeline
4
4.6 4.4
3.8
Malltown
4.2
4.5
A company wants to connect offices A, B, and C in
H 21. three
different cities with a telephone network. A, B,
and C form the vertices of an equilateral triangle.
The cities are about 400 miles from one another.
Should they connect A to B and B to C, or should
they install a phone at a fourth location (at the
center of triangle ABC) and connect it to A, B,
and C?
Overdeveloped
3.5
5
Nature’s Way
wants to connect offices A, B, C, and
H 22. AD company
in four different cities with a telephone network.
15. (a) What is a traversable circuit?
(b) What is an Euler circuit?
16. Are all traversable circuits also Euler circuits?
Explain why or why not.
A, B, C, and D form the vertices of a square. The
cities are about 300 miles from one another.
Should they connect A to B, B to C, and C to D, or
should they install a phone at a fifth location (at
the center of square ABCD) and connect it to A, B,
C, and D?
Extension Exercises
17. (1)
(2)
Find the area of a triangle with vertices (1, 4),
H 23. (3,
1), and (7, 2). (Hint: Enclose the triangle in a
(3)
rectangle.)
(a) Which of the networks are traversable?
(b) Which of the networks are Euler circuits?
18.
(1)
(2)
H 24. If (3, k) is equidistant from A(5, 3) and B(1, 5),
find the value of k.
is the set of points equidistant from a
H 25. Alineparabola
and a point not on that line. Follow steps (a)–(b)
(3)
to find the equation of the parabola that is equidistant from y 2 and (0, 4).
(a) Which of the networks are traversable?
(b) Which of the networks are Euler circuits?
y
(0, 4)
(x, y)
19. A warehouse has 3 work areas. The diagram shows
that there are 3 ways to go from area 1 to area 2 and
4 ways to go from area 2 to area 3. How many ways
are there to go from area 1 to area 3?
1
2
3
20. A warehouse has 3 work areas. The diagram shows
that there are 5 ways to go from area 1 to area 2 and
2 ways to go from area 2 to area 3. How many ways
are there to go from area 1 to area 3?
1
2
3
x
y = –2
(a) Any point (x, y) on the parabola is equidistant
from (0, 4) and y 2. What are the coordinates
of the point where y 2 intersects the perpendicular?
(b) Complete the following equation.
(0, 4) to (x, y) dist. (x, 2) to (x, y) dist.
( )2 ( )2 ( )2 ( )2
(c) Square both sides and combine like terms to
obtain the equation of the parabola.
(d) Does part (c) involve induction or deduction?
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11.6 Coordinate Geometry and Networks
26. Follow the instructions to create a curve by folding
waxed paper.
(a) Crease a line segment AB and mark a point C on
the waxed paper as shown.
.C
A
B
(b) Fold the point onto the segment and crease. Repeat this about 20 times, folding the point onto
different places on the segment.
(c) What shape is suggested by the creases?
(d) How does this work? (Hint: See the preceding
exercise.)
27. Find a point that is
15
3
of the way from (2, 5) to (4, 1).
4
28. The center of gravity of a polygon on a twodimensional graph is (x, y), in which x is the average
(mean) of the x-coordinates of the vertices and y is the
average (mean) of the y-coordinates of the vertices.
(a) Find the center of gravity of a quadrilateral with
vertices at coordinates A(1, 3), B(2, 4), C(5, 4),
and D(6, 3).
(b) Cut out a piece of cardboard that has the shape of
ABCD, and balance it on the tip of a pin. Is the
center of gravity very close to the location you
obtained in part (a)?
11.6 Answers to Selected Homework Exercises
1. Form a right triangle with C(x2, y1) as the third vertex. By the Pythagorean Theorem,
AB (AC)2 (CB)2 x
( 2 x1)2 (y2 y1)2
15. (a) A network with a path that passes through each
edge exactly one time
(b) A traversible network with a path that starts and
ends at the same place
3. 3 8 5 or 3 22 5
17. (a) (1), (2)
5. (1, 0), (1, 4), (3, 4)
19. 12
7. (3, 3)
21. Install a fourth location
9. (a) (a, b)
?
AC BC
(b)
D
?
(a 0)2 (b 0)2 (a 0)2 (0 b)2
a2 b2 a2 b2
23. 8 square units
(c) AC and BD bisect each other, since the midpoint
of AC is
a b
a b
,
and the midpoint of BD is , .
2 2
2 2
(b) (2)
25. (a) (x, 2)
(b) x2 (y 4)2 (0)2 (y 2)2
(c) x2 12y 12
(d) Deduction
27. (3.5, 2)
a b
ac b
11. (a) D( , ), E(
, )
2 2
2 2
kl
kl
(b) Slope of DE slope of AC 0
(a c)
2 a2 b2 b2
c
c
1
and AC c, so DE AC.
4
2
2
2
(c) DE 2
2
13. 32 min; 2
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