Special Right Triangles 8-2

8-2

8-2

Special Right Triangles

1. Plan

What You¡¯ll Learn

GO for Help

Check Skills You¡¯ll Need

? To use the properties of

1.

? To use the properties of

2.



2

308-608-908 triangles



2

4

2

2

To find the distance from

home plate to second base

on a softball diamond, as

in Example 3

3

45, 45, 90

2

3

4

5

Using 458-458-908 Triangles

=

Use the Pythagorean Theorem.

y2

x

45?

Simplify.

x !2 = y

Take the square root of each side.

x

45?

y

You have just proved the following theorem.

Key Concepts

Theorem 8-5

458-458-908 Triangle Theorem

In a 458-458-908 triangle, both legs are congruent and the

length of the hypotenuse is !2 times the length of a leg.

4

5

A

A

B

B

C

C

45?

s

45?

s

The ratio of the lengths of any

two sides of a right triangle is a

function of either acute angle.

This can be proved using similarity

theorems and is the basis for

the six trigonometric functions.

The fixed side-length ratios of

45¡ã-45¡ã-90¡ã and 30¡ã-60¡ã-90¡ã

triangles, easily found by applying

the Pythagorean Theorem,

provide benchmark values for

the trigonometric functions sine,

cosine, and tangent of 30¡ã, 45¡ã,

and 60¡ã angles.

More Math Background: p. 414C

E

E

D

C

B

E

D

C

B

A

3

s2

hypotenuse = !2 ? leg

E

D

C

B

A

D

C

B

A

Finding the Length of the

Hypotenuse

Finding the Length of a Leg

Real-World Connection

Using the Length of One Side

Real-World Connection

Math Background

x2 + x2 = y2

2x2

To use the properties of

45¡ã-45¡ã-90¡ã Triangles

To use the properties of

30¡ã-60¡ã-90¡ã Triangles

Examples

1

2

3

30, 60, 90

45, 45, 90

3

2

3

The acute angles of an isosceles right triangle are both 458 angles. Another name

for an isosceles right triangle is a 458-458-908 triangle. If each leg has length x and

the hypotenuse has length y, you can solve for y in terms of x.

2

1

2

3.

. . . And Why

1

Objectives

Use a protractor to ?nd the measures of the angles of each triangle.

458-458-908 triangles

1

Lesson 1-6

D

D

E

E

Test-Taking Tip

If you forget the

formula for a

45¡ã-45¡ã-90¡ã triangle,

use the Pythagorean

Theorem. The triangle

is isosceles, so the legs

have the same length.

1

Find the value of each variable.

a.

b.

9

45?

h = !2 ? 9

h = 9 !2

See p. 414E for a list of the

resources that support this lesson.

45?

x

45?

h

Quick Check

22

45?

PowerPoint

hypotenuse ¡Ù !2 ? leg

x = !2 ? 2 !2

Bell Ringer Practice

x=4

Simplify.

Check Skills You¡¯ll Need

1 Find the length of the hypotenuse of a 458-458-908 triangle with legs of length 5 !3.

5 "6

Lesson 8-2 Special Right Triangles

Special Needs

Lesson Planning and

Resources

Finding the Length of the Hypotenuse

EXAMPLE

Below Level

L1

For Example 3, have students check the answer by

cutting out a 60-mm by 60-mm square. They fold it

along its diagonal, and measure the length of the

diagonal to the nearest millimeter.

learning style: tactile

425

For intervention, direct students to:

Measuring Angles

Lesson 1-6: Example 2

Extra Skills, Word Problems, Proof

Practice, Ch. 1

L2

In the diagram for Theorem 8-6, construct a 30¡ã angle

adjacent to the 30¡ã angle, using a leg as one side.

Extend the base so that it intersects the new side. Discuss

why this forms an equilateral triangle.

learning style: visual

425

2. Teach

2

EXAMPLE

Finding the Length of a Leg

Multiple Choice What is the value of x?

3

Guided Instruction

1

EXAMPLE

6 = !2 ? x

x= 6

!2

x = 6 ? !2 = 6 !2

2

!2 !2

Technology Tip

Point out that using mental

math is much faster than using a

calculator for part b. The calculator

answer also would be inexact,

whereas squaring the square root

of a number is always exact.

2

EXAMPLE

x = 3 !2

Quick Check

Real-World

Connection

Careers Opportunities for

coaching in women¡¯s sports

have soared since the passage

of Title IX in 1972.

3 The distance from one corner

to the opposite corner of a square

playground is 96 ft. To the nearest

foot, how long is each side of the

playground? 68 ft

Quick Check

2

1

Suggest that students distinguish

between the 45¡ã-45¡ã-90¡ã and the

30¡ã-60¡ã-90¡ã Triangle Theorems by

using the ¡°ratio¡± diagrams below.

2

1 45¡ã

45¡ã

1

Divide each side by !2.

45?

Multiply by a form of 1.

Simplify.

EXAMPLE

Real-World

Connection

The distance d from home plate to second

base is the length of the hypotenuse of a

458-458-908 triangle.

d = 60 !2

d=

84.852814

hypotenuse ¡Ù !2 ? leg

60 ft

d

Use a calculator.

On a high school softball diamond, the catcher throws the ball about 85 ft from

home plate to second base.

3 A square garden has sides 100 ft long. You want to build a brick path along a

diagonal of the square. How long will the path be? Round your answer to the

nearest foot. 141 ft

Another type of special right triangle is a 308-608-908 triangle.

Key Concepts

Theorem 8-6

308-608-908 Triangle Theorem

In a 308-608-908 triangle, the length of the hypotenuse

is twice the length of the shorter leg. The length of the

longer leg is !3 times the length of the shorter leg.

60¡ã

1

2s 30? s

3

hypotenuse = 2 ? shorter leg

60?

s

longer leg = !3 ? shorter leg

Error Prevention!

426

Chapter 8 Right Triangles and Trigonometry

Advanced Learners

English Language Learners ELL

L4

After students learn and apply Theorem 8-5, have

them write a formula for the area of an isosceles right

triangle whose hypotenuse has length s.

426

6

Using 308-608-908 Triangles

3 30¡ã 2

Whenever the length of a

hypotenuse or longer leg of a

30¡ã-60¡ã-90¡ã triangle is given,

encourage students to find the

length of the shorter leg first.

45?

Softball A high school softball diamond is a square. The distance from base to

base is 60 ft. To the nearest foot, how far does a catcher throw the ball from home

plate to second base?

Additional Examples

Visual Learners

x

2 Find the length of a leg of a 458-458-908 triangle with a hypotenuse of length 10.

5 "2

3

PowerPoint

of a 45¡ã-45¡ã-90¡ã triangle with

a hypotenuse of length 22. 11"2

6 !2

When you apply the 458-458-908 Triangle Theorem to a real-life example, you can

use a calculator to evaluate square roots.

Have several students explain

aloud to the class how to

rationalize a denominator.

2 Find the length of a leg

6

hypotenuse ¡Ù !2 ? leg

The correct answer is B.

Auditory Learners

1 Find the length of the

hypotenuse of a 45¡ã-45¡ã-90¡ã

triangle with legs of length 5"6.

10"3

3 !2

learning style: verbal

Ask students to complete each statement: The

shortest side of a triangle is always opposite the

smallest angle. In a 30¡ã-60¡ã-90¡ã triangle, the shortest

side is always opposite the 30¡ã angle .

learning style: verbal

4

To prove Theorem 8-6, draw a 308-608-908 triangle using an equilateral triangle.

Proof

Proof of Theorem 8-6

XY 2 + YW 2 = XW 2

s 2 + YW 2 = (2s)2

Use the Pythagorean Theorem.

Subtract s2 from each side.

YW 2 = 3s 2

Simplify.

YW = s !3

PowerPoint

30?

60?

s

X

Additional Examples

60?

Y

Z

Substitute s for XY and 2s for XW.

YW 2 = 4s 2 - s 2

Find the square root of each side.

The 30¡ã-60¡ã-90¡ã Triangle Theorem, like the 45¡ã-45¡ã-90¡ã Triangle Theorem, lets you

?nd two sides of a triangle when you know the length of the third side.

4

EXAMPLE

Using the Length of One Side

Algebra Find the value of each variable.

5 = d !3

d=

5 ? !3 = 5 !3

3

"3 !3

f = 2d

10 !3

f = 2 ? 5 !3

3 = 3

Quick Check

longer leg ¡Ù ? 3 ? shorter leg

5

EXAMPLE

60?

30?

Closure

Substitute 5 !3

3 for d.

8

30?

60?

s ¡Ö 1.155

x

y

C

75¡ã

B

s

A

1m

The triangle is equilateral, so the altitude divides the

triangle into two 30¡ã-60¡ã-90¡ã triangles as shown in the

diagram. The altitude also bisects the base, so the shorter

leg of each 30¡ã-60¡ã-90¡ã triangle is 12 s.

2 5s

!3

In quadrilateral ABCD, AD = DC

and AC = 20. Find the area of

ABCD. Leave your answer in

simplest radical form.

Connection

Road Signs The moose warning sign at the left is an

equilateral triangle. The height of the sign is 1 m. Find

the length s of each side of the sign to the nearest tenth

of a meter.

1 = !3 Q 12 sR

5 A garden shaped like a

rhombus has a perimeter of 100 ft

and a 60¡ã angle. Find the

perpendicular height between two

bases. 21.7 ft

f

hypotenuse ¡Ù 2 ? shorter leg

Real-World

4 The longer leg of a 30¡ã-60¡ã-90¡ã

triangle has length 18. Find the

lengths of the shorter leg and the

hypotenuse. shorter leg: 6"3;

hypotenuse: 12"3

Resources

? Daily Notetaking Guide 8-2 L3

? Daily Notetaking Guide 8-2¡ª

L1

Adapted Instruction

5

d

Solve for d.

4 Find the value of each variable. x = 4, y = 4"3

Math Tip

Students can use the Pythagorean

Theorem to check their work.

W

For 308-608-908 #WXY in equilateral #WXZ,

WY is the perpendicular bisector of XZ.

Thus, XY = 12 XZ = 12 XW, or XW = 2XY = 2s.

Also,

EXAMPLE

105¡ã

D

100 ¡À 50"3

60?

1s

2

longer leg ¡Ù ? 3 ? shorter leg

Solve for s.

Simplify. Use a calculator.

Each side of the sign is about 1.2 m long.

Quick Check

5 If the sides of the sign are 1 m long, what is the height? about 0.9 m

Lesson 8-2 Special Right Triangles

427

427

EXERCISES

3. Practice

Assignment Guide

A

1 A B 1-8, 21, 22, 26

2 A B

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

Practice and Problem Solving

Practice by Example

Example 1

9-20, 23-25

C Challenge

27-28

Test Prep

Mixed Review

29-32

33-41

GO for

Help

(page 425)

Examples 2 and 3

Find the value of each variable. If your answer is not an integer, leave it in simplest

radical form.

x ¡Ù "2; y ¡Ù 2

y ¡Ù 60 "2

x ¡Ù 8; y ¡Ù 8 "2 2.

1.

3.

y 45? x

45? y

y

8

60

45?

x

45?

2

Homework Quick Check

x ¡Ù 15; y ¡Ù 15

4.

(page 426)

45?

152

y

To check students¡¯ understanding

of key skills and concepts, go over

Exercises 4, 12, 18, 24, 25.

5.

Exercises 17¨C19 Students should

first find any side length that can

be derived using a given side.

After the first length is found,

the other lengths often fall

into place.

9.

10.

40

x

60?

y

L4

L2

L1

Practice

Name

Class

L3

Date

Practice 8-2

Similar Polygons

Are the polygons similar? If they are, write a similarity statement, and give

the similarity ratio. If they are not, explain.

1.

2.

X

A

12

10

R

Q

6

5

7

Y

20

¡ª

3

Z

20

¡ª

3

A

3.

4

4

S 3

T

5

B

14

B

4.

J

8

C

M

K

120?

8

8

M

8

60?

8

R

B

16

A

7. ?M  ?

9

9

21

8. ?K  ?

6

L

6

M

4

4

N

N

M

W

4

Z

B

J

9. ?N  ?

K

N

M

IJ = HK

12. MN

?

HI

11. HK

= LM

?

18. a ¡Ù 6; b ¡Ù 6 "2;

c ¡Ù 2"3; d ¡Ù 6

19. a ¡Ù 10 "3; b ¡Ù 5 "3;

c ¡Ù 15; d ¡Ù 5

Z

6. X 4 Y

K

17. a ¡Ù 7; b ¡Ù 14; c ¡Ù 7;

d ¡Ù 7 "3

35

20

LMNO M HIJK. Complete the proportions and congruence statements.

?

10. MN

IJ = JK

5

C

N

C

U

5

3

3

K

12

8

120?

60?

L

5.

T

8

S

8

5

X

3

Y

L

O

H

C

14. P

A

6 ft

L

B

E

x

3 in.

F

S

G

15.

16. J

x

X

W

K

x

H

F

M

Y

1.5 cm

Z

4 cm

R

8 in.

15 m

L

x

O

P

6m

S 9m

N

Q

R

30?

d

E

G

kWXZ M kDFG. Use the diagram to ?nd the following.

18. m?Z

19. DG

20. GF

21. m?G

22. m?D

23. WZ

428

b

a

10 cm

Z

17. the similarity ratio of ?WXZ and ?DFG

3

W

37?

4

X

D

60?

x

x ¡Ù 5; y ¡Ù 5 "3

3

9

x

60?

y

x ¡Ù 9; y ¡Ù 18

25 ft

30?

16. City Planning Jefferson Park sits on one square city block 300 ft on each side.

Sidewalks join opposite corners. About how long is each diagonal sidewalk?

424 ft

17¨C19. See above left.

Apply Your Skills x 2 Algebra Find the value of each variable. Leave your answer in simplest radical form.

2

7

45?

c

M

5 in.

3.3 ft

14.

60?

30?

x

y

x ¡Ù 4; y ¡Ù 2

x ¡Ù 24; y ¡Ù 12 "3

15. Architecture An escalator lifts people to

the second ?oor, 25 ft above the ?rst ?oor.

The escalator rises at a 30¡ã angle.

How far does a person travel from the

bottom to the top of the escalator? 43 ft

17.

Q

G

5 ft

12

10

y

3

2

y

30?

I

Algebra The polygons are similar. Find the values of the variables.

13.

13.

60?

x

L3

30?

3

2

x ¡Ù "3; y ¡Ù 3

x ¡Ù 20; y ¡Ù 20 "3

12.

11.

y

x

30?

Adapted Practice

5

8. Helicopters The four blades of a helicopter meet at right angles and are

all the same length. The distance between the tips of two adjacent blades

Exercise 7

is 36 ft. How long is each blade? Round your answer to the nearest tenth.

25.5 ft

Examples 4 and 5 x 2 Algebra Find the value of each variable. If your answer is not an integer, leave it in

simplest radical form.

(page 427)

Exercises 20¨C22 Each of these

exercises requires constructing

an altitude to form a rectangle.

Reteaching

y

5

7. Dinnerware Design You are designing dinnerware. What is the length of

a side of the smallest square plate on which a 20-cm chopstick can ?t along

a diagonal without any overhang? Round your answer to the nearest tenth

of a centimeter. 14.1 cm

bring chopsticks to class and

demonstrate how to use them.

Enrichment

"10

6.

45?

x

45?

x

Exercise 7 Ask a volunteer to

GPS Guided Problem Solving

8 4 "2

6

F

428

Chapter 8 Right Triangles and Trigonometry

18.

19.

3

4

60?

c

a

b

45?

d

a

30?

c

10

60?

b

d

GO

6

20.

nline

Homework Help

Visit:

Web Code: aue-0802

23. Rika; Sandra marked

the shorter leg as

opposite the 608 angle.

3

2

60?

a

4

21.

32

a

4. Assess & Reteach

8

22.

b

45?

6

45?

PowerPoint

a

b

b

a ¡Ù 3; b ¡Ù 7

a ¡Ù 4; b ¡Ù 4

23. Error Analysis Sandra drew the triangle at

the right. Rika said that the lengths couldn¡¯t

be correct. With which student do you agree?

Explain your answer.

Lesson Quiz

a ¡Ù 14; b ¡Ù 6 "2

Use kABC for Exercises 1¨C3.

5

A

5 3

60?

30?

10

24. Open-Ended Write a real-life problem that

you can solve using a 308-608-908 triangle with

a 12 ft hypotenuse. Show your solution. See margin.

25. Farming A conveyor belt carries bales of hay from the ground to the barn loft

GPS 24 ft above the ground. The belt makes a 608 angle with the ground.

a. How far does a bale of hay travel from one end of the conveyor belt to the

other? Round your answer to the nearest foot. 28 ft

b. The conveyor belt moves at 100 ft/min. How long does it take for a bale of

hay to go from the ground to the barn loft? 0.28 min

Exercise 25

C

Challenge

27a. "3 units

b. 2 "3 units

c. s"3 units

26. House Repair After heavy winds damaged a farmhouse,

workers placed a 6-m brace against its side at a 458 angle.

Then, at the same spot on the ground, they placed a second,

longer brace to make a 308 angle with the side of the house.

a. How long is the longer brace? Round your answer to the

nearest tenth of a meter. 8.5 m

b. How much higher on the house does the longer brace

reach than the shorter brace? 3.1 m

27. Geometry in 3 Dimensions Find the length

d, in simplest radical form, of the diagonal

of a cube with sides of the given length. See left.

a. 1 unit

b. 2 units

c. s units

28. Constructions Construct a 30¡ã-60¡ã-90¡ã triangle

given a segment that is

a. the shorter leg.

b. the hypotenuse.

c. the longer leg. See back of book.

30?

6m

45?

1

1

1

18

B

1. If m&A = 45, find AC and AB.

AC ¡Ù 18; AB ¡Ù 18"2

2. If m&A = 30, find AC and AB.

AC ¡Ù 18"3; AB ¡Ù 36

3. If m&A = 60, find AC

and AB. AC ¡Ù 6"3;

AB ¡Ù 12"3

2

d

C

4. Find the side length of a

45¡ã-45¡ã-90¡ã triangle with a

4-cm hypotenuse. 2"2 N

2.8 cm

5. Two 12-mm sides of a triangle

form a 120¡ã angle. Find the

length of the third side.

12"3 N 20.8 mm

d

2

2

Alternative Assessment

Have students use compass and

straightedge to construct a large

equilateral triangle with one

altitude. Then have them explain

how the three sides of one of the

right triangles are related.

Test Prep

Test Prep

Multiple Choice

29. What is the length of a diagonal of a square with sides of length 4? D

A. 2

B. !2

C. 2 !2

D. 4 !2

Resources

30. The longer leg of a 30¡ã-60¡ã-90¡ã triangle is 6. What is the length of the

hypotenuse? H

A. 2 !3

B. 3 !2

C. 4 !3

D. 12

31. The hypotenuse of a 30¡ã-60¡ã-90¡ã triangle is 30. What is the length of one of

its legs? D

A. 3 !10

B. 10 !3

C. 15 !2

D. 15

lesson quiz, , Web Code: aua-0802

Lesson 8-2 Special Right Triangles

For additional practice with a

variety of test item formats:

? Standardized Test Prep, p. 465

? Test-Taking Strategies, p. 460

? Test-Taking Strategies with

Transparencies

429

24. Answers may vary.

Sample: A ramp up to a

door is 12 ft long. It has

an incline of 30¡ã. How

high off the ground is the

door? sol.: 6 ft

429

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