CorrectionKey=NL-A;CA-A Name Class Date 8.1 … · Module 8 359 Lesson 1 8.1 Perpendicular Bisecors...

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© Houghton Mifflin Harcourt Publishing Company Name Class Date Explore Constructing a Circumscribed Circle A circle that contains all the vertices of a polygon is circumscribed about the polygon. In the figure, circle C is circumscribed about XYZ, and circle C is called the circumcircle of XYZ. The center of the circumcircle is called the circumcenter of the triangle. In the following activity, you will construct the circumcircle of PQR. Copy the triangle onto a separate piece of paper. A The circumcircle will pass through P, Q, and R. So, the center of the circle must be equidistant from all three points. In particular, the center must be equidistant from Q and R. The set of points that are equidistant from Q and R is called the of _ QR . Use a compass and straightedge to construct the set of points. B The center must also be equidistant from P and R. The set of points that are equidistant from P and R is called the of _ PR . Use a compass and straightedge to construct the set of points. C The center must lie at the intersection of the two sets of points you constructed. Label the point C. Then place the point of your compass at C and open it to distance CP. Draw the circumcircle. Resource Locker C X Z Y C R P Q Module 8 359 Lesson 1 8.1 Perpendicular Bisectors of Triangles Essential Question: How can you use perpendicular bisectors to find the point that is equidistant from all the vertices of a triangle?

Transcript of CorrectionKey=NL-A;CA-A Name Class Date 8.1 … · Module 8 359 Lesson 1 8.1 Perpendicular Bisecors...

Page 1: CorrectionKey=NL-A;CA-A Name Class Date 8.1 … · Module 8 359 Lesson 1 8.1 Perpendicular Bisecors t of Triangles Essential Question: How can you use perpendicular bisectors to find

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Name Class Date

Explore Constructing a Circumscribed CircleA circle that contains all the vertices of a polygon is circumscribed about the polygon. In the figure, circle C is circumscribed about △XYZ, and circle C is called the circumcircle of △XYZ. The center of the circumcircle is called the circumcenter of the triangle.

In the following activity, you will construct the circumcircle of △PQR. Copy the triangle onto a separate piece of paper.

A The circumcircle will pass through P, Q, and R. So, the center of the circle must be equidistant from all three points. In particular, the center must be equidistant from Q and R.

The set of points that are equidistant from Q and R is called the of

_ QR .

Use a compass and straightedge to construct the set of points.

B The center must also be equidistant from P and R. The set of points that are equidistant from P and R is called the of

_ PR . Use a compass and

straightedge to construct the set of points.

CThe center must lie at the intersection of the two sets of points you constructed. Label the point C. Then place the point of your compass at C and open it to distance CP. Draw the circumcircle.

Resource Locker

CXZ

Y

CR

P

Q

Module 8 359 Lesson 1

8.1 Perpendicular Bisectors of Triangles

Essential Question: How can you use perpendicular bisectors to find the point that is equidistant from all the vertices of a triangle?

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Reflect

1. Make a Prediction Suppose you started by constructing the set of points equidistant from P and Q and then constructed the set of points equidistant from Q and R. Would you have found the same center? Check by doing this construction.

2. Can you locate the circumcenter of a triangle without using a compass and straightedge? Explain.

Explain 1 Proving the Concurrency of a Triangle’s Perpendicular Bisectors

Three or more lines are concurrent if they intersect at the same point. The point of intersection is called the point of concurrency. You saw in the Explore that the three perpendicular bisectors of a triangle are concurrent. Now you will prove that the point of concurrency is the circumcenter of the triangle. That is, the point of concurrency is equidistant from the vertices of the triangle.

Circumcenter Theorem

The perpendicular bisectors of the sides of a triangle intersect at a point that is equidistant from the vertices of the triangle.

PA = PB = PC A

PC

B

Example 1 Prove the Circumcenter Theorem.

Given: Lines ℓ, m, and n are the perpendicular bisectors of _ AB , _ BC , and

_ AC ,

respectively. P is the intersection of ℓ, m, and n.

Prove: PA = PB = PC

P is the intersection of ℓ, m, and n. Since P lies on the

of _ AB , PA = PB by the Theorem. Similarly, P lies on

the of _ BC , so = PC. Therefore, PA = =

by the Property of Equality.

C

A

P

B

m

nℓ

Module 8 360 Lesson 1

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Reflect

3. Discussion How might you determine whether the circumcenter of a triangle is always inside the triangle? Make a plan and then determine whether the circumcenter is always inside the triangle.

Explain 2 Using Properties of Perpendicular BisectorsYou can use the Circumcenter Theorem to find segment lengths in a triangle.

Example 2 _ KZ , _ LZ , and

_ MZ are the perpendicular bisectors of △GHJ. Use the

given information to find the length of each segment. Note that the figure is not drawn to scale.

A Given: ZM = 7, ZJ = 25, HK = 20

Find: ZH and HG

Z is the circumcenter of △GHJ, so ZG = ZH = ZJ.

ZJ = 25, so ZH = 25.

K is the midpoint of _ GH , so HG = 2 ⋅ KH = 2 ⋅ 20 = 40.

B Given: ZH = 85, MZ = 13, HG = 136

Find: KG and ZJ

K is the of _ HG , so KG = HG = · = .

Z is the of △GHJ, so ZG = = .

ZH = , so ZJ = .

G

K L

JM

Z

H

Module 8 361 Lesson 1

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Reflect

4. In △ABC, ∠ACB is a right angle and D is the circumcenter of the triangle. If CD = 6.5, what is AB? Explain your reasoning.

Your Turn

KZ , LZ , and MZ are the perpendicular bisectors of △GHJ. Copy the sketch and label the given information. Use that information to find the length of each segment. Note that the figure is not drawn to scale.

5. Given: ZG = 65, HL = 63, ZL = 16 Find: GK and ZJ

6. Given: ZM = 25, ZH = 65, GJ = 120 Find: GM and ZG

C

BD

A

G

K L

JM

Z

H

Module 8 362 Lesson 1

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Explain 3 Finding a Circumcenter on a Coordinate Plane

Given the vertices of a triangle, you can graph the triangle and use the graph to find the circumcenter of the triangle.

Example 3 Graph the triangle with the given vertices and find the circumcenter of the triangle.

A R ( -6, 0 ) , S ( 0, 4 ) , O ( 0, 0 )

Step 1: Graph the triangle.

Step 2: Find equations for two perpendicular bisectors.

Side _ RO is on the x-axis, so its perpendicular bisector is vertical:

the line x = -3.

Side _ SO is on the y-axis, so its perpendicular bisector

is horizontal: the line y = 2.

Step 3: Find the intersection of the perpendicular bisectors.

The lines x = -3 and y = 2 intersect at (-3, 2) .

(-3, 2) is the circumcenter of △ROS.

B A (-1, 5) , B (5, 5) , C (5, -1)

Step 1 Graph the triangle.

Step 2 Find equations for two perpendicular bisectors.

Side _ AB is , so its perpendicular bisector

is vertical

The perpendicular bisector of _ AB is the line. .

Side _ BC is , so the perpendicular bisector of

_ BC is horizontal the line .

Step 3 Find the intersection of the perpendicular bisectors.

The lines and intersect at .

is the circumcenter of △ABC.

y

0

6

2

R O

S

x

x = -3

y = 2 (-3, 2)

-2

-4

y

0-2

5

3

1

7

-2642

x

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Reflect

7. Draw Conclusions Could a vertex of a triangle also be its circumcenter? If so, provide an example. If not, explain why not.

Your Turn

Graph the triangle with the given vertices and find the circumcenter of the triangle.

8. Q (-4, 0) , R (0, 0) , S (0, 6) 9. K (1, 1) , L (1, 7) , M (6, 1)

Elaborate

10. A company that makes and sells bicycles has its largest stores in three cities. The company wants to build a new factory that is equidistant from each of the stores. Given a map, how could you identify the location for the new factory?

11. A sculptor builds a mobile in which a triangle rotates around its circumcenter. Each vertex traces the shape of a circle as it rotates. What circle does it trace? Explain.

y

0-2-4-6

2

4

6

-22

x

y

0-2

2

4

6

-22 4 6

x

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12. What If? Suppose you are given the vertices of a triangle PQR. You plot the points in a coordinate plane and notice that

_ PQ is horizontal but neither of the other sides is vertical. How can you identify the

circumcenter of the triangle? Justify your reasoning.

13. Essential Question Check-In How is the point that is equidistant from the three vertices of a triangle related to the circumcircle of the triangle?

Construct the circumcircle of each triangle. Label the circumcenter P.

1. 2.

3. 4.

• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

A

CB

A

BC

B

CA

C

B

A

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Complete the proof of the Circumcenter Theorem.

Use the diagram for Exercise 5–8. ZD, ZE, and ZF are the perpendicular bisectors of △ABC. Use the given information to find the length of each segment. Note that the figure is not drawn to scale.

5. Given: DZ = 40, ZA = 85, FC = 77

Find: ZC and AC

6. Given: FZ = 36, ZA = 85, AB = 150

Find: AD and ZB

7. Given: AZ = 85, ZE = 51

Find: BC

(Hint: Use the Pythagorean Theorem.)

8. Analyze Relationships How can you write an algebraic expression for the radius of the circumcircle of △ABC in Exercises 6–8? Explain.

A

FZ

CE

B

D

Module 8 366 Lesson 1

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Complete the proof of the Circumcenter Theorem.

9. Given: Lines ℓ, m, and n are the perpendicular bisectors of _ AB , _ BC , and

_ AC ,

respectively. P is the intersection of ℓ, m, and n.

Prove: PA = PB = PC

Statements Reasons

1. Lines ℓ, m, and n are the perpendicular bisectors of

_ AB , _ BC , and

_ AC .

1.

2. P is the intersection of ℓ, m, and n. 2.

3. PA = 3. P lies on the perpendicular bisector of _ AB .

4. = PC 4. P lies on the perpendicular bisector of _ BC .

5. PA = = 5.

10. _ PK , _ PL , and

_ PM are the perpendicular bisectors of sides

_ AB , _ BC ,

and _ AC . Tell whether the given statement is justified by the figure.

Select the correct answer for each lettered part.a. AK = KB Justified Not Justifiedb. PA = PB Justified Not Justifiedc. PM = PL Justified Not Justifiedd. BL = 1 _ 2 BC Justified Not Justifiede. PK = KD Justified Not Justified

C

A

P

B

m

nℓ

P

M D C

B

LK

A

Module 8 367 Lesson 1

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Graph the triangle with the given vertices and find the circumcenter of the triangle.

11. D (-5, 0) , E (0, 0) , F (0, 7)

12. Q (3, 4) , R (7, 4) , S (3, -2)

13. Represent Real-World Problems For the next Fourth of July, the towns of Ashton, Bradford, and Clearview will launch a fireworks display from a boat in the lake. Draw a sketch to show where the boat should be positioned so that it is the same distance from all three towns. Justify your sketch.

H.O.T. Focus on Higher Order Thinking

14. Analyze Relationships Explain how can you draw a triangle JKL whose circumcircle has a radius of 8 centimeters.

y

0-2-4-6

2

4

6

x

y

0

2

4

2 4 6

x

-2

Ashton

Clearview

Bradford

Final art 3/15/05ge07se_c05l02005aGeometry SE 2007 TexasHolt Rinehart WinstonKaren Minot(415)883-6560

Final art 3/15/05 ge07se_c05l02006a Geometry SE 2007 Texas Holt Rinehart Winston Karen Minot (415)883-6560

A

C

B

F

Module 8 368 Lesson 1

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15. Persevere in Problem Solving _ ZD , _ ZE and

_ ZF are the perpendicular bisectors of

△ABC, which is not drawn to scale.

a. Suppose that ZB = 145, ZD = 100, and ZF = 17. How can you find AB and AC?

b. Find AB and AC.

c. Can you find BC? If so, explain how and find BC. If not, explain why not.

16. Multiple Representations Given the vertices A (-2, -2) , B (4, 0) , and C (4, 4) of a triangle, the graph shows how you can use a graph and construction to locate the circumcenter P of the triangle. You can draw the perpendicular bisector of

_ CB and construct the

perpendicular bisector of _ AB . Consider how you could identify

P algebraically.

a. The perpendicular bisector of _ AB passes through its midpoint.

Use the Midpoint Formula to find the midpoint of _ AB .

b. What is the slope m of the perpendicular bisector of _ AB ? Explain

how you found it.

c. Write an equation of the perpendicular bisector of _ AB and explain how

you can use it find P.

A

FZ

CE

B

D

y

0-2-4

4

6

-2

-4

642

x

C (4, 4)

B (4, 0)

A (-2, -2)

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Lesson Performance TaskA landscape architect wants to plant a circle of flowers around a triangular garden. She has sketched the triangle on a coordinate grid with vertices at A (0, 0) , B (8, 12) , and C (18, 0) .

Explain how the architect can find the center of the circle that will circumscribe triangle ABC. Then find the radius of the circumscribed circle.

y

A (0, 0) C (18, 0)

B (8, 12)

x

4 8 12 16

4

8

12

16

Module 8 370 Lesson 1

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