46 ft46 ft

DD

CC

BB

AA

EE

Using a Midsegment of a Triangle

Environmental Science A geologist wants to determine the distance, AB, across a sinkhole. Choosing a point E outside the sinkhole, she finds the distances AE and BE. She locates the midpoints C and D of AE and BE and then measures CD. What is the distance across the sinkhole?

CD is a midsegment of △AEB.

CD = 12 AB △ Midsegment Thm.

46 = 12 AB Substitute 46 for CD.

92 = AB Multiply each side by 2.

The distance across the sinkhole is 92 ft.

TEKS Process Standard (1)(A)

STEM

Problem 4

PRACTICE and APPLICATION EXERCISES

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1. Explain Mathematical Ideas (1)(G) The triangular face of the Rock and Roll Hall of Fame in Cleveland, Ohio, is isosceles. The length of the base is 229 ft 6 in. Each leg is divided into four congruent parts by the red segments. What is the length of the white segment? Explain your reasoning.

2. Use Representations to Communicate Mathematical Ideas (1)(E) You want to paddle your kayak across a lake. To determine how far you must paddle, you pace out a triangle, counting the number of strides, as shown. If your strides average 3.5 ft, what distance must you paddle across the lake?

80

80

150

150

250

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Why does the geologist find the length of CD?CD is a midsegment of △AEB, so the geologist can use its length to find AB, the distance across the sinkhole.

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3. Analyze Mathematical Relationships (1)(F) In the figure at the right, m∠QST = 40. What is m∠QPR? Explain how you know.

4. Draw △ABC. Construct another triangle so that the three sides of △ABC are the midsegments of the new triangle.

5. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) The coordinates of the vertices of a triangle are E(1, 2), F(5, 6), and G(3, -2). Let H be the midpoint of EG and let J be the midpoint of FG. Verify the Triangle Midsegment Theorem by showing that HJ } EF and HJ = 1

2 EF .

IJ is a midsegment of △FGH, where IJ is parallel to FG. IJ = 7, FH = 10, and GH = 13. Find the perimeter of each triangle.

6. △IJH 7. △FGH

Points E, D, and H are the midpoints of the sides of △TUV. UV = 80, TV = 100, and HD = 80.

8. Find HE. 9. Find ED.

10. Find TU. 11. Find TE.

Find the value of x.

12. 13. 14.

15. Analyze Mathematical Relationships (1)(F) In △GHJ, K(2, 3) is the midpoint of GH, L(4, 1) is the midpoint of HJ, and M(6, 2) is the midpoint of GJ . Find the coordinates of G, H, and J.

16. Complete the Prove statement and then write a proof.

Given: In △VYZ , S, T, and U are midpoints.

Prove: △YST ≅ △TUZ ≅ △SVU ≅ ?

Verify the Triangle Midsegment Theorem for each triangle.

17. △FGH with vertices F(–6, 4), G(4, 8), and H(2, –2), where J and K are the midpoints of FG and FH

18. △PQR with vertices P(–4, –8), Q(–10, 0), and R(0, 0), where S and T are the midpoints of RP and RQ

X is the midpoint of UV. Y is the midpoint of UW.

19. If m∠UXY = 60, find m∠V.

20. If XY = 50, find VW.

21. If VW = 110, find XY.

S T

R

Q

P U

T

H

V D U

E

26

x5x ! 4

8 72

6x

ProofY

V U Z

TS

U

X Y

WV

204 Lesson 5-2 Midsegments of Triangles

Mmmmm

22. Apply Mathematics (1)(A) You design a kite to look like the one at the right. Its diagonals measure 64 cm and 90 cm. You plan to use ribbon, represented by the purple rectangle, to connect the midpoints of its sides. How much ribbon do you need?

A. 77 cm C. 154 cm

B. 122 cm D. 308 cm

Use the figure at the right.

23. DF = 24, BC = 6, and DB = 8. Find the perimeter of △ADF .

24. If BE = 2x + 6 and DF = 5x + 9, find DF.

25. If EC = 3x - 1 and AD = 5x + 7, find EC.

26. Explain Mathematical Ideas (1)(G) Explain how you could use the Triangle Midsegment Theorem as the basis for this construction: Draw CD. Draw point A not on CD. Construct AB so that AB } CD and AB = 1

2 CD.

27. Use a Problem-Solving Model (1)(B) In the diagram at the right, K, L, and M are the midpoints of the sides of △ABC. The vertices of the three small purple triangles are the midpoints of the sides of △KBL, △AKM , and △MLC. The perimeter of △ABC is 24 cm. What is the perimeter of the shaded region?

28. Connect Mathematical Ideas (1)(F) Explain how verifying the Triangle Midsegment Theorem is different from proving the Triangle Midsegment Theorem.

CB

D

FEA

B

K

A M C

L

TEXAS Test Practice

Use the figure at the right for Exercises 29 and 30. Your home is at point H. Your friend lives at point F, the midpoint of Elm Street. Elm Street intersects Beech Street and Maple Street at their midpoints.

29. Your friend walks to school by going east on Elm and then turning right on Maple. How far in miles does she walk?

30. You walk your dog along this route: Walk from home to Elm along Maple. Walk west on Elm to Beech, south on Beech to the library, and east on Oak to school. Then walk back home along Maple. How far in miles do you walk?

For Exercises 31 and 32, △ABC is a triangle in which mjA = 30 and mjB = 70. P, Q, and R are the midpoints of AB, BC, and CA, respectively.

31. What is the measure, in degrees, of ∠RPQ?

32. If QP = 2x + 17 and CA = x + 97, what is CA?

H

F

N

Library Oak St. School

Elm St.

Beec

h St

. Maple St.

3.2 mi

1 mi

1.2 mi

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Brimmed

9. Explain Mathematical Ideas (1)(G) You and a friend compete in a scavenger hunt at a museum. The two of you walk from the Picasso exhibit to the Native American gallery along the dashed red line. When he sees that another team is ahead of you, your friend says, “They must have cut through the courtyard.” Explain what your friend means.

10. Select Techniques to Solve Problems (1)(C) Your family drives across Kansas on Interstate 70. A sign reads, “Wichita 90 mi, Topeka 110 mi.” Your little brother says, “I didn’t know that it was only 20 miles from Wichita to Topeka.” Explain why the distance between the two cities does not have to be 20 mi.

Can a triangle have sides with the given lengths? Explain.

11. 2 in., 3 in., 6 in. 12. 11 cm, 12 cm, 15 cm 13. 2 yd, 9 yd, 10 yd

Analyze Mathematical Relationships (1)(F) The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side.

14. 5 in., 16 in. 15. 18 m, 23 m 16. 20 km, 35 km

17. Apply Mathematics (1)(A) You are setting up a study area where you will do your homework each evening. It is triangular with an entrance on one side. You want to put your computer in the corner with the largest angle and a bookshelf on the longest side. Where should you place your computer? On which side should you place the bookshelf? Explain.

List the angles of each triangle in order from smallest to largest.

18. 19.

For Exercises 20–23, list the sides of each triangle in order from shortest to longest.

20. 21.

22. △ABC, with 23. △XYZ , with m∠A = 90, m∠X = 51, m∠B = 40, and m∠Y = 59, and m∠C = 50 m∠Z = 70

Native American Gallery

PicassoExhibit

7 ft

Entrance

C

E

D

105!3x

x JG

4

6

H

G

F

H28!

110!

T

U V30!

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Write an inequality relating the given side lengths. If there is not enough information to reach a conclusion, write no conclusion.

1. AB and AD 2. PR and RT

3. Justify Mathematical Arguments (1)(G) Complete the following proof.

Given: C is the midpoint of BD, m∠EAC = m∠AEC, m∠BCA 7 m∠DCE

Prove: AB 7 ED

Statements Reasons1) m∠EAC = m∠AEC 1) Given

2) AC = EC 2) a. ?

3) C is the midpoint of BD. 3) b. ?

4) BC ≅ CD 4) c. ?

5) d. ? 5) ≅ segments have = length.

6) m∠BCA 7 m∠DCE 6) e. ?

7) AB 7 ED 7) f. ?

Analyze Mathematical Relationships (1)(F) Find the range of possible values for each variable.

4. 5.

6. The diagram below shows a robotic arm in two different positions. In which position is the tip of the robotic arm closer to the base? Use the Hinge Theorem to justify your answer.

60! 40!

A

B

C

D

38!

41!

Q

P

S

T

R62! 67!

B

E

A

D

C

7 8

32!(x " 6)! 5 636!

(2x " 12)!

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238 Lesson 5-8 Inequalities in Two Triangles

7. Use the figure at the right.

Given: △ABE is isosceles with vertex ∠B, △ABE ≅ △CBD, m∠EBD 7 m∠ABE

Prove: ED 7 AE

8. Apply Mathematics (1)(A) Ship A and Ship B leave from the same point in the ocean. Ship A travels 150 mi due west, turns 65° toward north, and then travels another 100 mi. Ship B travels 150 mi due east, turns 70° toward south, and then travels another 100 mi. Which ship is farther from the starting point? Explain.

9. Justify Mathematical Arguments (1)(G) △ABC has vertices A(0, 7), B(-1, -2), and C(2, -1). For O(0, 0), show that m∠AOB 7 m∠AOC.

10. Which of the following lists the segment lengths in order from least to greatest?

A. CD, AB, DE, BC, EF

B. EF, DE, AB, BC, CD

C. BC, DE, EF, AB, CD

D. EF, BC, DE, AB, CD

ProofA

B

C

DE

Start

Ship A

Ship B

70!

100 mi

100 mi

150 mi150 mi

65!

N

W E

S

77!

74!

76!73! 75!

A O

B

CD

E

F

TEXAS Test Practice

11. A quilter cuts out and sews together two triangles, as shown in the figure. She tries to make △ABD and △BDC equilateral triangles, but the angle measures are as shown. Assuming that the side BD is actually the same length in both triangles, which segment is the longest side in the figure?

F. AB H. CD

G. BC J. AD

12. Use indirect reasoning to write a convincing argument that a triangle has at most one obtuse angle.

13. Which is a possible value for x in the figure at the right?

A. 2 C. 6

B. 4 D. 8

59! C

B

A59!61!

61!60! 60!

D

38! 43!

2x " 7x

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mmmm M