11.6 Coordinate Geometry and Networks - · PDF file8 Chapter 11Algebra and Graphing ... 11.6 Coordinate Geometry and Networks ... use geometric models to gain insights into algebra

8 Chapter 11 Algebra and Graphing

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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 regularpolygons or those with pairs of parallel or perpendicular sides (6–8)

• use visual tools such as networks to represent and solve problems (6–8)

H

B(x2, y2)

CA(x1, y1)

Figure 11–21

Coordinate geometry establishes a connection between algebra and geometry. Thisenables mathematicians to employ algebraic methods to solve geometry problems and touse geometric models to gain insights into algebra problems.

The Distance FormulaHow 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 SkillA seventh grader computes DE in LE 1 as . Is this right? Ifnot, what would you tell the child?

The Pythagorean Theorem is the basis for computing the distance between any twopoints on a two-dimensional graph. Complete the following exercise to derive the dis-tance formula.

LE 3 Reasoning(a) For two points A(x1, y1) and B(x2, y2), construct a right triangle with hypotenuse

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?

AB

�32 � 22 � 3 � 2 � 5

61668_11_ch11_sec11.6.qxd 4/22/09 11:08 AM Page 8

If the slope of AB is negative, the derivation is similar and the resulting formula isthe same. The distance formula tells us how to find the length of a line segment usingthe coordinates of its endpoints.

11.6 Coordinate Geometry and Networks 9


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 ConnectionA quadrilateral ABCD has coordinates A(0, 0), B(5, 0), C(8, 4), and D(3, 4). Use thedistance formula to show that ABCD is a rhombus.

The Midpoint FormulaHow are the coordinates of the midpoint of a line segment related to the coordinates of theendpoints?

LE 5 Reasoning(a) Plot the points A(3, 2), B(�1, 4), and C(5, 6), and draw �ABC on graph paper.(b) Find the coordinates of the midpoint of .(c) Find the coordinates of the midpoint of .(d) Find the coordinates of the midpoint of .(e) 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 ?(g) Using parts (b)–(d) to answer parts (e) and (f) is an example of what kind of

reasoning?

The answer to LE 5(f) is the midpoint formula.

AB

BCABAC

H

The Midpoint Formula

Given A(x1, y1) and B(x2, y2), then the midpoint M of is

�x1 � x2

2,

y1 � y2

2 �AB

The midpoint and distance formulas can be used to find properties of triangles andquadrilaterals.

61668_11_ch11_sec11.6.qxd 4/22/09 11:08 AM Page 9



LE 6 Reasoning�ABC is a right triangle with A(0, 0), B(8, 0), and C(0, 6).

(a) Plot �ABC on a coordinate graph.(b) Find the coordinates of M, the midpoint of the hypotenuse.(c) Compare the lengths AM, BM, and CM.(d) Repeat parts (a)–(c) with a new right triangle.(e) Make a generalization based on your results.

In Chapter 8, you studied properties of parallelograms. The following exerciseshows how to use coordinate geometry to prove one such property: that the diagonals ofa parallelogram bisect each other.

LE 7 Reasoning

(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 is the midpoint of .

NetworksSuppose 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 cansolve this problem by drawing a graph of the four cities (see Figure 11–23) called a net-work. A network is a graph whose edges are labeled with numbers such as lengths ortimes.

LE 8 ReasoningFind the fastest route that starts in Chicago and goes through all 3 cities and then backto Chicago.

LE 9 ReasoningA phone company wants to link up 4 cities (Figure 11–24). The network shows thecost (in millions of dollars) of different possible links. Find the cheapest possiblenetwork.

The points in a network are called vertices, and the curves are called edges. Anetwork 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 networktraversable?

BDAC

H

H

x

y

C (a + b, c)D (b, c)

B (a, 0)A (0, 0)

Chicago Detroit

PittsburghSt. Louis

3.5

12 hr

6 hr5.5 hr 10 10

Allview

Cool City Dullsville

Boomtown3

2.12.5

3.42.8 2.7

Figure 11–22

Figure 11–23

Figure 11–24

H

H

61668_11_ch11_sec11.6.qxd 4/22/09 11:08 AM Page 10

LE 10 Reasoning

(a) Which of the following networks are traversable? Mark where you started andwhere you finished.

(b) Determine whether the vertices of each network are odd or even. An even numberof 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 anetwork traversable.

To determine if a network is traversable, check whether each vertex is even or odd.

A business sometimes desires a traversable network where one can start and end inthe same place.

LE 11 ReasoningA street cleaner wants to find a route that travels along each street exactly once andstarts and ends at the same place. Such a route is called an Euler circuit. What prop-erty of the vertices makes a network an Euler circuit?

Since an Euler circuit is traversable and has the same starting and ending point, itmust have all even vertices.

LE 12 Reasoning

(a) Tell which of the following has an Euler circuit from point A.

(b) Would it be possible to make an Euler circuit from a different starting point ineither network?

Route 1

A

Route 2

A

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 theother will be where you finish.

(1)

(4)

(2) (3)

(5)



H

H

H

61668_11_ch11_sec11.6.qxd 4/22/09 11:08 AM Page 11



LE 13 Reasoning

(a) Which of the networks are traversable?(b) Which of the networks are Euler circuits?

(1) (2) (3)

H

1. (b) 2; 3 (c)

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)(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)

(g) Inductive

6. (b) M(4, 3) (c) AM � BM � CM � 5(e) The midpoint of the hypotenuse is equidistant

from the three vertices.

�x1 � x2

2,

y1 � y2

2 �

�(x2 � x1)2 � (y2 � y1)

2

�13 7. (b) Hint: Compute slopes to show that the oppositesides are parallel.

(c) The midpoints of both diagonals are .

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)

�a � b2

, c2�

Answers to Selected Lesson Exercises

LE 14 Summary

(a) Tell what you learned about coordinate geometry in this section. How is thedistance formula related to the Pythagorean Theorem?

(b) Tell what you learned about networks. What are traversable networks and Eulercircuits?

61668_11_ch11_sec11.6.qxd 4/22/09 11:08 AM Page 12

H

H

H

H

H

H

H



11.6 Homework Exercises

Basic Exercises1. Explain why the distance between A(x1, y1) and

B(x2, y2) is

2. Explain how the distance formula is related to thePythagorean Theorem.

3. Find the perimeter of �ABC with coordinates A(3, 6), B(3, 9), and C(5, 7).

4. Determine whether or not (0, 4), (4, 5), and (6, �3)are the vertices of a right triangle.

5. Three vertices of a parallelogram are (1, �2), (0, 2),and (2, 0). Find three possible coordinates for thefourth vertex.

6. Three vertices of a rhombus are (1, 2), (4, 1), and (3, �2). What are the coordinates of the fourth vertex?

7. Point A has coordinates (4, �1), and point B hascoordinates (2, 7). What are the coordinates of themid-point of ?

8. Point M is the midpoint of . Point A has coordi-nates (3, 7), and M has coordinates (�2, 1). Find thecoordinates of B.

9. ABCD is a rectangle.

(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.

x

y

B (a, 0)A (0, 0)

CD (0, b)

AB

AB

x

y B (x2, y2)

A (x1, y1)

AB � �(x2 � x1)2 � (y2 � y1)

2

10.

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.

11.

(a) Find the coordinates of the midpoints D of and E of .

(b) Show that .(c) How does DE compare to AC? Prove it.

12.

(a) Find the coordinates of the midpoints of the foursides, 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. Whatis the fastest route he could take? How many differ-ent routes of the minimum time are there?

Home Park

Store Library

5 min.

10 min.105

1210

x

y

A

H

D(0, b) G C

F

B (a, 0)E

DE ´ ACBC

AB

x

y

C (c, 0)

A

B (a, b)

x

y

B (a, 0)A

CD (0, a)

61668_11_ch11_sec11.6.qxd 4/22/09 11:08 AM Page 13

H

H

H

H

H



14. A phone company wants to link up 5 cities. Thenetwork shows the cost (in millions of dollars) ofdifferent possible links. Find the cheapest possiblenetwork.

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.

17.


18.


19. A warehouse has 3 work areas. The diagram showsthat 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 waysare there to go from area 1 to area 3?

20. A warehouse has 3 work areas. The diagram showsthat 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 waysare there to go from area 1 to area 3?

1 2 3

1 2 3

(1) (2) (3)

(1) (2) (3)

Quagmire

Nature’s Way

OverdevelopedMalltown

Pipeline4

4.2

3.55

3.8

4.4

4.5

4.6

21. A company wants to connect offices A, B, and C inthree 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 shouldthey install a phone at a fourth location (at thecenter of triangle ABC) and connect it to A, B, and C?

22. A company wants to connect offices A, B, C, and D in four different cities with a telephone network.A, B, C, and D form the vertices of a square. Thecities are about 300 miles from one another.Should they connect A to B, B to C, and C to D, orshould they install a phone at a fifth location (atthe center of square ABCD) and connect it to A, B,C, and D?

Extension Exercises23. Find the area of a triangle with vertices (�1, 4),

(3, 1), and (7, 2). (Hint: Enclose the triangle in a rectangle.)

24. If (3, k) is equidistant from A(5, �3) and B(�1, �5),find the value of k.

25. A parabola is the set of points equidistant from aline and a point not on that line. Follow steps (a)–(b)to find the equation of the parabola that is equidis-tant from y � �2 and (0, 4).

(a) Any point (x, y) on the parabola is equidistantfrom (0, 4) and y � �2. What are the coordinatesof the point where y � �2 intersects the perpen-dicular?

(b) Complete the following equation.

(0, 4) to (x, y) dist. � (x, �2) to (x, y) dist.

(c) Square both sides and combine like terms toobtain the equation of the parabola.

(d) Does part (c) involve induction or deduction?

�( )2 � ( )2 � �( )2 � ( )2

(x, y)

(0, 4)

x

y

y = –2

61668_11_ch11_sec11.6.qxd 4/22/09 11:08 AM Page 14



26. Follow the instructions to create a curve by foldingwaxed paper.(a) Crease a line segment and mark a point C on

the waxed paper as shown.

(b) Fold the point onto the segment and crease. Re-peat this about 20 times, folding the point ontodifferent places on the segment.

(c) What shape is suggested by the creases?(d) How does this work? (Hint: See the preceding

exercise.)

A B.C

AB

27. Find a point that is of the way from (2, 5) to (4, 1).

28. The center of gravity of a polygon on a two-dimensional graph is ( ), in which is the average(mean) of the x-coordinates of the vertices and is theaverage (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 ofABCD, and balance it on the tip of a pin. Is thecenter of gravity very close to the location youobtained in part (a)?

yxx, y

34

11.6 Answers to Selected Homework Exercises

1. Form a right triangle with C(x2, y1) as the third ver-tex. By the Pythagorean Theorem,

3. or

5. (�1, 0), (1, 4), (3, �4)

7. (3, 3)

9. (a) (a, b)(b)

(c) and bisect each other, since the midpoint

of is and the midpoint of is .

11. (a) D( , ), E( , )

(b) Slope of � slope of

(c)

13. 32 min; 2

� �c2

4 � c

2 and AC � c, so DE � 1

2 AC.

DE � ��(a � c)2

� a2�

2

� �b2

� b2�

2

k

ACl

� 0k

DEl

b2

a � c2

b2

a2

�a2

, b2�BD�a

2, b

2�AC

BDAC

�a2 � b2 � �a2 � b2

�(a � 0)2 � (b � 0)2 �?

�(a � 0)2 � (0 � b)2

AC �?

BC

3 � 2�2 � �53 � �8 � �5

AB � �(AC)2 � (CB)2 � �x2 � x1)2 � (y2 � y1)

2

15. (a) A network with a path that passes through eachedge exactly one time

(b) A traversible network with a path that starts andends at the same place

17. (a) (1), (2) (b) (2)

19. 12

21. Install a fourth location

23. 8 square units

25. (a) (x, �2)

(b)

(c) x2 � 12y � 12 (d) Deduction

27. (3.5, 2)

�x2 � (y � 4)2 � �(0)2 � (y � 2)2

D

(

61668_11_ch11_sec11.6.qxd 4/22/09 11:09 AM Page 15

Documents

11.6 Coordinate Geometry and Networks - · PDF file8 Chapter 11Algebra and Graphing ... 11.6 Coordinate Geometry and Networks ... use geometric models to gain insights into algebra