5-1 Midsegments of Triangles

[Pages:6]5-1

Midsegments of Triangles

Lesson Preview

What You'll Learn

? To use properties of

midsegments to solve problems

. . . And Why

To use indirect measurement to find the length of a lake, as in Example 3

Check Skills You'll Need

GO for Help Lesson 1-8 and page 165

Find the coordinates of the midpoint of each segment.

1. AB with A(-2, 3) and B(4, 1) (1, 2)

2.

CD with C(0, 5) and D(3, 6)

A32

,

11 2

B

3. EF with E(-4, 6) and F(3, 10) A?12, 8B

4. GH with G(7, 10) and H(-5, -8) (1, 1)

Find the slope of the line containing each pair of points.

5. A(-2, 3) and B(3, 1) ?52

7.

E(-4, 6) and F(3, 10)

4 7

6.

C(0, 5) and D(3, 6)

1 3

8.

G(7, 10) and H(-5, -8)

3 2

New Vocabulary ? midsegment ? coordinate proof

1 Using Properties of Midsegments

2. Answers may vary. Sample: The midsegment is n to the 3rd side of the k and is half its length.

Hands-On Activity: Midsegments of Triangles

Draw, label, and cut out a large scalene

triangle. Do the same with other right,

acute, and obtuse triangles. Label the

vertices A, B, and C.

A

? For each triangle fold A onto C to find the midpoint of AC. Do the same for BC. Label the midpoints L and N, then draw LN.

L A

C

B C

N B

? Fold each triangle on LN.

1.

LN

1 2

AB;

Explanations

may

vary.

? Fold A to C. Fold B to C.

1. How does LN compare to AB? Explain.

A

B

B

2. Make a conjecture about how the segment joining the midpoints of

two sides of a triangle is related to the third side of the triangle. See left.

In #ABC above, LN is a triangle midsegment. A midsegment of a triangle is a segment connecting the midpoints of two sides.

Lesson 5-1 Midsegments of Triangles 259

5-1

1. Plan

Objectives

1 To use properties of midsegments to solve problems

Examples

1 Finding Lengths 2 Identifying Parallel Segments 3 Real-World Connection

Math Background

Euclid did not use coordinate geometry to prove any theorems. The Triangle Midsegment Theorem can be proved without coordinate geometry, but the proof requires theorems concerning parallelograms that are not presented in this text until Chapter 6.

More Math Background: p. 256C

Lesson Planning and Resources

See p. 256E for a list of the resources that support this lesson.

PowerPoint

Bell Ringer Practice

Check Skills You'll Need For intervention, direct students to:

Finding the Midpoint of a Segment Lesson 1-8: Example 3 Extra Skills, Word Problems, Proof

Practice, Ch. 1 Slope Algebra Review, p. 165

Special Needs L1 In the Hands-On Activity, some students may not see why folding A onto C marks the midpoint of AC because of its orientation. Demonstrate by folding the corners of a rectangular piece of paper.

learning style: visual

Below Level L2 To eliminate the fractions in proving Theorem 5-1, let the respective coordinates of points Q and P be (2a, 0) and (2b, 2c).

learning style: verbal

259

2. Teach

Guided Instruction

Hands-On Activity

Have students place labels for the vertices inside the triangle on both sides of the paper so they will appear on the cut-out figures. Instruct students to label the obtuse or right angle vertex C to ensure that the first folded triangle lies inside ABC.

Connection to Algebra

The proof of the Triangle Midsegment Theorem uses the Midpoint and Distance Formulas from Chapter 1 and the calculation of slope from Chapter 3. Ask: Why are variables used in the proof instead of numbers? Using numbers proves the theorem for one set of points. Because any number can be substituted for a variable, using variables proves the theorem for all sets of points.

Math Tip

Discuss as a class why the vertices in the proof of the Triangle Midsegment Theorem are labeled O(0, 0), Q(a, 0), and P(b, c). Explain that translating, rotating, or reflecting a triangle so that two of its vertices are at (0, 0) and (a, 0) simplifies using the Midpoint and Distance Formulas.

1 EXAMPLE Auditory Learners

Have students read through Example 1 in small groups. Then ask volunteers to explain how the example applies the Triangle Midsegment Theorem.

Key Concepts

Theorem 5-1

Triangle Midsegment Theorem

If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length.

One way to prove the Triangle Midsegment Theorem is to use coordinate geometry and algebra. This style of proof is called a coordinate proof. You begin the proof by placing a triangle in a convenient spot on the coordinate plane. You then choose variables for the coordinates of the vertices.

Proof

Vocabulary Tip

The Midpoint Formula

Qx1

1 2

x2,

y1

1 2

y2 R

The Distance Formula

$(x2 2 x1)2 1 ( y2 2 y1)2

Proof of Theorem 5-1

Given: R is the midpoint of OP.

S is the midpoint of QP.

Prove:

RS

6

OQ

and

RS

=

1 2

OQ

? Use the Midpoint Formula to find the

coordinates of R and S.

R:

Q0

1 2

b,

0

1 2

cR

=

Qb2,

c 2

R

S:

Qa

1 2

b,

0

1 2

cR

=

Qa

1 2

b,

c 2

R

y P (b, c)

R O(0, 0)

S

x Q(a, 0)

? To prove that RS and OQ are parallel, show that their slopes are equal. Because the y-coordinates of R and S are the same, the slope of RS is zero. The same is true for OQ. Therefore, RS 6 OQ.

? Use the Distance Formula to find RS and OQ.

RS

=

%Qa

1 2

b

2

b 2

R

2

1

Q2c

2

2c R2

=

%Qa2

1

b 2

2

b2 R2

1

02

=

% Q a2 R 2

=

a 2

=

1 2

a

OQ = $(a 2 0)2 1 (0 2 0)2

= $a2 1 02 = a

Therefore,

RS

=

1 2

OQ.

1 EXAMPLE Finding Lengths

In #EFG, H, J, and K are midpoints. Find HJ, JK, and FG.

HJ

=

1 2

EG

or

12(100);

HJ

=

50

JK

=

1 2

EF

or

12(60);

JK

=

30

HK

or

40

=

1 2

FG;

FG

=

80

F

60 H

J

40

E

K

G

100

Quick Check 1 AB = 10 and CD = 18. Find EB, BC, and AC.

EB 9; BC 10; AC 20

A

E

B

D

C

260 Chapter 5 Relationships Within Triangles

260

Advanced Learners L4 Have students use Theorem 5-1 to prove that the midpoints of three sides of a triangle can be used to form four congruent triangles.

learning style: verbal

English Language Learners ELL Help students relate Example 3 with the Triangle Midsegment Theorem. Ask questions such as: Why did Dean not just measure the distance across the lake? Why did he mark 35 paces on one side and 118 paces on the other side?

learning style: verbal

2 EXAMPLE Identifying Parallel Segments

In #DEF, A, B, and C are midpoints. Name pairs of parallel segments.

The midsegments are AB, BC, and CA.

E

A

B

By the Triangle Midsegment Theorem, AB 6 DF, BC 6 ED, and AC 6 EF

D C

F

Quick Check 2 Critical Thinking Find m&VUZ. Justify your answer.

X

65; UV n XY so lVUZ and lYXZ are corr. and O.

65

U

You can use the Triangle Midsegment Theorem

Y

to find lengths of segments that might be difficult

V

Z

to measure directly.

3 EXAMPLE Real-World Connection

Indirect Measurement Dean plans to swim the length of the lake, as shown in the photo. How far would Dean swim?

Here is what Dean does to find the distance he would swim across the lake.

Step 1: He measures his stride and adjusts it so that it averages about 3 ft.

Step 2: Then he begins at the left edge of the lake (first diagram). He paces 35 strides along the edge of the lake and sets a stake.

Step 3: He paces 35 more strides in

the same direction and sets

35

another stake. 35

Step 4: He paces to where his swim

will end at the other side of

the lake, counting 236 strides.

236 ?

Step 5: Then (second diagram) he paces 118 strides, or half the distance, back towards the second stake.

118 128

Step 6: He paces to the first stake,

?

counting 128 strides.

Step 7: He converts strides to feet.

128

strides

3

1

3 ft stride

=

384

ft

Step 8: He uses Theorem 5-1. The distance across the lake is twice the length of the midsegment.

2(384 ft) = 768 ft

Dean would swim approximately 768 ft.

Quick Check

3 a. CD is a new bridge being built over a lake

as shown. Find the length of the bridge. 1320 ft

b. How long is the bridge in miles?

1 4

mi

963 ft C

963 ft

Bridge

2640 ft

D

Lesson 5-1 Midsegments of Triangles 261

Connection to Astronomy

Astronomers use indirect measurement to measure great distances. Have students research how astronomers measure distances in the universe.

PowerPoint

Additional Examples

1 In XYZ, M, N, and P are midpoints. The perimeter of MNP is 60. Find NP and YZ.

X

M 22

P

24

Y

N

Z

NP 14; YZ 44

2 Find m&AMN and m&ANM. A

N

M

C

75? B

mlAMN mlANM 75

3 Explain why Dean could use the Triangle Midsegment Theorem to measure the length of the lake. He paced between the midpoints of two sides of a triangle.

Resources

? Daily Notetaking Guide 5-1 L3

? Daily Notetaking Guide 5-1--

Adapted Instruction

L1

Closure

The perimeter of a triangle is 78 ft. Find the perimeter of the triangle formed by its midsegments. 39 ft

261

3. Practice

Assignment Guide

1 A B 1-36 C Challenge

37-39

Test Prep Mixed Review

40-46 47-55

Homework Quick Check

To check students' understanding of key skills and concepts, go over Exercises 3, 20, 21, 26, 30.

Error Prevention!

Exercise 13 Students may misapply the Triangle Midsegment Theorem, thinking that the angles are also in a 1 : 2 ratio. Review the theorem with the class before beginning this exercise.

Exercise 29 After students solve the exercise, show them how they can directly find the answer by simply adding the measures of the diagonals. This follows because each side of the ribbon is half of a diagonal.

GPS Guided Problem Solving

L3

Enrichment

Reteaching

Adapted Practice

PraNcamte ice

Class

Practice 5-1

Use the diagrams at the right to complete the exercises.

1. In MNO, the points C, D, and E are midpoints. CD = 4 cm, CE = 8 cm, and DE = 7 cm.

a. Find MO.

b. Find NO.

c. Find MN.

2. In quadrilateral WVUT, the points F, E, D, and C are midpoints. WU = 45 in. and TV = 31 in.

a. Find CD.

b. Find CF.

c. Find ED.

3. In LOB, the points A, R, and T are midpoints. LB = 19 cm, LO = 35 cm, and OB = 29 cm.

a. Find RT.

b. Find AT.

c. Find AR.

Find the value of the variable. 4. x 34

5. y

41

L4

L2

L1

Date

L3

Midsegments of Triangles

N

C

D

M

O

E

VEU

F

D

W CT

R

O

L

A

T

B

6.

7 2t

? Pearson Education, Inc. All rights reserved.

7. Perimeter of ABC = 32 cm

A

n

12? n

B

78? n

C

8.

q 21

9.

t 33 37

10. QR is a midsegment of LMN.

L

a. QR = 9. Find NM. b. LN = 12 and LM = 31. Find the perimeter of LMN.

Q

R

N

M

Use the given measures to identify three pairs of parallel segments in each diagram.

11.

B

4

G 4

A

6

7

H 7

I

6

C

12. X

Y

P Q

R

Z

EXERCISES

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

Practice and Problem Solving

A Practice by Example

GO

for Help

Example 1 (page 260)

Mental Math Find the value of x.

1. 9

2.

7

5x x

70 18

4.

x -1

45

2312

5. 11 x 1 5

3. 14

84 3x

6. 2

5x - 2 4

Example 2 (page 261)

Example 3 (page 261) 20a. 1050 ft

Points E, D, and H are midpoints of kTUV. UV 80, TV 100, and HD 80.

7. Find HE. 40

8. Find ED. 50

T

H

E

9. Find TU. 160 10. Find TE. 80

V

D

U

Identify pairs of parallel segments in each diagram. GJ n FK; JL n HF; GL n HK

11. UW n TX; UY n VX;

V

YW n TV

U

W

T

Y

X

12.

H

56

G

J

5

6

F 7L7 K

13. a. In the figure at the right, identify

pairs of parallel segments.ST n PR; SU n QR; UT n PQ

b. If m&QST = 40, find m&QPR. mlQPR 40

S

Q T

P

Name the segment that is parallel to the given segment.

14. AB FE

15. BC FG

A

16. EF AB

17. CA EG

F

18. GE AC

19. FG CB

C

U

G E

R B

20. Indirect Measurement Kate wants to paddle her canoe across the lake. To determine how 80 far she must paddle, she paced out a triangle, counting the number of strides, as shown. a. If Kate's strides average 3.5 ft, what is the 80 length of the longest side of the triangle? b. What distance must Kate paddle across the lake? 437.5 ft

150

150 250

262 Chapter 5 Relationships Within Triangles

262

B Apply Your Skills

Problem Solving Hint

The highlighted segment is halfway up the face of the Rock and Roll Hall of Fame.

21b. Answers may vary. Sample: The highlighted segment is a midsegment of the triangular face of the building.

21. a. Architecture 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. What is the length of the highlighted segment? 114 ft 9 in.

b. Writing Explain your reasoning. See left.

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

22. If m&UXY = 60, find m&V. 60

23. If m&W = 45 find m&UYX. 45

24. If XY = 50, find VW. 100

25. If VW = 110, find XY. 55

U X V

Y W

26. Coordinate Geometry The coordinates of the vertices of a triangle are E(1, 2),

F(5, 6), and G(3, -2). a. H(2, 0); J(4, 2) b-c. See margin.

a. Find the coordinates of H, the midpoint of EG, and J, the midpoint of FG.

b. Verify that HJ 6 EF.

c. Verify that HJ = 12EF.

H

IJ is a midsegment of #FGH. IJ = 7, FH = 10, and

GH = 13. Find the perimeter of each triangle.

I

J

27. #IJH 1812

28. #FGH 37

F

G

29. Multiple Choice Marita is designing a kite to look like the one on the left. Its

diagonals are to measure 64 cm and 90 cm. She will use ribbon to connect the

midpoints of its sides. How much ribbon will Marita need? C

77 cm

122 cm

154 cm

308 cm

Exercise 29

x2 Algebra Find the value of each variable.

30. GPS

30 60

31.

50

x

21

25 x

32. 10

x 5

60

33.

3x 6

y

x 2x 1

x

6;

y

6

1 2

GO nline

Use the figure at the right for Exercises 34?36. D

Homework Help

34. If DF = 24, BC = 6, and DB = 8,

Visit:

find the perimeter of #ADF. 52

B

C

Web Code: aue-0501

x2 35. Algebra If BE = 2x + 6 and DF = 5x + 9,

find the value of x, then find DF. x 3; DF 24 A

E

F

x2 36. Algebra If EC = 3x - 1 and AD = 5x + 7,

find the value of x, then find EC. x 9; EC 26

lesson quiz, , Web Code: aua-0501

26.

b.

Slope

of

HJ

2 2

1;

slope

of

EF

4 4

1;

therefore HJ n EF.

c. HJ "22 1 22

"8 2"2; EF

Lesson 5-1 Midsegments of Triangles 263

"42 1 42 "32

4"2;

therefore

HJ

1 2

EF.

4. Assess & Reteach

PowerPoint

Lesson Quiz

In kGHI, R, S, and T are midpoints.

H

R

S

G

T

I

RT n HI, RS n GI, ST n HG 1. Name all the pairs of parallel

sides of GHI and RST.

2. If GH = 20 and HI = 18, find RT. 9

3. If RH = 7 and RS = 5, find ST. 7

4. If m&G = 60 and m&I = 70, find m>R. 70

5. If m&H = 50 and m&I = 66, find m&ITS. 64

6. If m&G = m&H = m&I and RT

= 15, find the perimeter of GHI. 90

Alternative Assessment

Draw the figure below on the board. Label the vertices of the large triangle and the midpoints of the sides. Name the triangles and the midsegments.

D

70?

A

c

B

b

a

60?

F

C

50? E

Have students use the given information to find the lengths of the sides of DEF and the measures of the angles of ABC. Then have students explain in writing how they found the measures of the sides and angles.

263

Test Prep

A sheet of blank grids is available in the Test-Taking Strategies with Transparencies booklet. Give this sheet to students for practice with filling in the grids.

Resources For additional practice with a variety of test item formats: ? Standardized Test Prep, p. 301 ? Test-Taking Strategies, p. 296 ? Test-Taking Strategies with

Transparencies

37. Answers may vary.

Sample: Draw CA and S

extend CA to P so that

CA AP. Find B, the midpt. of PD. Then, by

the k Midsegment Thm.,

AB

n

CD

and

AB

1 2

CD.

39. kUTS; Proofs may vary.

Sample: VS O SY, YT O TZ, and VU O UZ

because S, T, and U are

midpts. of the respective

sides;

ST

1 2

VZ

so

ST O VU O UZ;

SU

1 2

YZ

so

SU

O

YT

O

TZ;

and

TU

1 2

VY

so TU O SY O SV;

therefore kYST O kTUZ

O kSVU O kUTS by SSS.

50.

y

(0, 2) 2

4 2 O 2 4

2 4x

yx?2

51.

y

4

2

4 2 O (0, 2) 2

2 4x

y 3x ? 2

264

C Challenge

37. Open-Ended 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 6 CD and AB

=

1 2

CD.

See

margin.

38. Coordinate Geometry 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. G(4, 4); H(0, 2); J(8, 0)

Proof 39. Complete the prove statement and then write a proof.

Given: S, T, and U are midpoints.

Y

Prove: #YST > #TUZ > #SVU > 9.

See margin.

S

T

V

U

Z

Test Prep

Gridded Response

Q and P are midpoints of two sides of #RST.

40. What is RS? 248 41. What is TQ? 174 42. What is TS? 418

R x + 50

P x

x + 85

Q

S 3x + 46 T

43. What is m&ABC? 70 44. What is m&D? 40 45. What is m&A? 70 46. What is m&CBE? 40

A

B

1408 C

F 708

E

BE 6 AD

not to scale D

Mixed Review

Lesson 4-7

GO

for Help

Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL.

47.

ST

48. A

E

49. L

N

W

B

D

X

Y

kSXT O kTYS; SAS

C kADC O kEBC; ASA

Lesson 3-6 x2 Algebra Graph each line. 50?52. See margin.

50. y = x + 2

51. y = 3x - 2

M

KR

QP

kKLQ O kPNR; HL

52. y = -x - 5

Lesson 3-2 x2 Algebra Determine the value of x for which < n m.

53.

(3x + 4) 54.

144

m

46

2 3

35

70 (2x)

m

264 Chapter 5 Relationships Within Triangles

55. (3x - 5)

115 m

40

52.

y

2

y ?x ? 5

2 O 2

2 4x

(0, 5) 6

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