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Lesson 8-2 Special Right Triangles 425
Special Right Triangles
Lesson 1-
Use a protractor to find the measures of the angles of each triangle.
1. 2. 3.
What You’ll Learn
- To use the properties of 458 -45 8 -90 8 triangles
- To use the properties of 308 -60 8 -90 8 triangles
... And Why
To find the distance from home plate to second base on a softball diamond, as in Example 3
The acute angles of an isosceles right triangle are both 45 8 angles. Another name for an isosceles right triangle is a 45 8 -45 8 -90 8 triangle. If each leg has length x and the hypotenuse has length y , you can solve for y in terms of x.
x^2 + x^2 = y^2 Use the Pythagorean Theorem. 2 x^2 = y^2 Simplify. x = y Take the square root of each side. You have just proved the following theorem.
Finding the Length of the Hypotenuse
Find the value of each variable. a. b.
h =? 9 hypotenuse ≠? leg x =? 2 h = 9 Simplify. x = 4
Quick Check (^11) Find the length of the hypotenuse of a 45 8 -45 8 -90 8 triangle with legs of length 5! 3.
x 45
h
11 EXAMPLE^ EXAMPLE
x
y
x 45 45
11 Using 45^8 -45^8 -90^8 Triangles
Key Concepts Theorem 8-5^458 -45^8 -90^8 Triangle Theorem
In a 45 8 -45 8 -90 8 triangle, both legs are congruent and the length of the hypotenuse is times the length of a leg. hypotenuse =! 2? leg
! 2 s
s
s 2 45
Check Skills You’ll Need (^) GO for Help
Test-Taking Tip
1 2 A^ A BB^ C^ C D^ D EE 3 4 AA^ B^ B C^ C DD^ EE 5 A^ B^ B C^ C D^ D EE
If you forget the formula for a 45°-45°-90° triangle, use the Pythagorean Theorem. The triangle is isosceles, so the legs have the same length.
PowerPoint
Special Needs
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.
Below Level
In the diagram for Theorem 8-6, construct a 30° angle adjacent to the 30° angle, using a leg as one side. Extend the base so that it intersects the new side. Discuss why this forms an equilateral triangle.
L1 L
learning style: tactile learning style: visual
1. Plan
Objectives
1 To use the properties of 45°-45°-90° Triangles 2 To use the properties of 30°-60°-90° Triangles
Examples
1 Finding the Length of the Hypotenuse 2 Finding the Length of a Leg 3 Real-World Connection 4 Using the Length of One Side 5 Real-World Connection
Math Background
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°-45°-90° and 30°-60°-90° triangles, easily found by applying the Pythagorean Theorem, provide benchmark values for the trigonometric functions sine, cosine, and tangent of 30°, 45°, and 60° angles.
More Math Background: p. 414C
Lesson Planning and Resources
See p. 414E for a list of the resources that support this lesson.
Bell Ringer Practice
Check Skills You’ll Need
For intervention, direct students to:
Measuring Angles
Lesson 1-6: Example 2 Extra Skills, Word Problems, Proof Practice, Ch. 1
Finding the Length of a Leg
Multiple Choice What is the value of x? 3 3 6 6 6 =? x hypotenuse ≠? leg x = Divide each side by.
x =? = Multiply by a form of 1. x = 3 Simplify. The correct answer is B.
Find the length of a leg of a 45 8 -45 8 -90 8 triangle with a hypotenuse of length 10.
When you apply the 45 8 -45 8 -90 8 Triangle Theorem to a real-life example, you can use a calculator to evaluate square roots.
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? The distance d from home plate to second base is the length of the hypotenuse of a 458 -45 8 -90 8 triangle.
d = 60 hypotenuse ≠? leg d = (^) 84.852814 Use a calculator.
On a high school softball diamond, the catcher throws the ball about 85 ft from home plate to second base.
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 30 8 -60 8 -90 8 triangle.
Quick Check^33
d
60 ft
33 EXAMPLE^ EXAMPLE^ Real-World^ Connection
Quick Check 22
6! 2 2
! 2 ! 2
6 ! 2
!^62!^2
x
45
22 EXAMPLE^ EXAMPLE
Careers Opportunities for coaching in women’s sports have soared since the passage of Title IX in 1972.
(^12) Using 30 8 -60 8 -90 8 Triangles
Key Concepts Theorem 8-6^308 -60^8 -90^8 Triangle Theorem
In a 30 8 -60 8 -90 8 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is times the length of the shorter leg. hypotenuse = 2? shorter leg
longer leg =! 3? shorter leg
! 3 2 s
s
s 3
Real-World Connection
426 Chapter 8 Right Triangles and Trigonometry
- Teach
Guided Instruction
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.
Auditory Learners
Have several students explain aloud to the class how to rationalize a denominator.
Additional Examples
Find the length of the hypotenuse of a 45°-45°-90° triangle with legs of length 5.
10
Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22. 11
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
Visual Learners
Suggest that students distinguish between the 45°-45°-90° and the 30°-60°-90° Triangle Theorems by using the “ratio” diagrams below.
Error Prevention!
Whenever the length of a hypotenuse or longer leg of a 30°-60°-90° triangle is given, encourage students to find the length of the shorter leg first.
1
3 2
60 °
^30 °
1
2 1 45 °
45 °
33
22
11
22 EXAMPLEEXAMPLE
11 EXAMPLEEXAMPLE
Advanced Learners
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.
English Language Learners ELL
Ask students to complete each statement: The shortest side of a triangle is always opposite the smallest angle. In a 30°-60°-90° triangle, the shortest side is always opposite the 30° angle.
L
learning style: verbal learning style: verbal
PowerPoint
Find the value of each variable. If your answer is not an integer, leave it in simplest radical form.
1. 2. 3.
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 fit along a diagonal without any overhang? Round your answer to the nearest tenth of a centimeter. 14.1 cm 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 is 36 ft. How long is each blade? Round your answer to the nearest tenth.
Algebra Find the value of each variable. If your answer is not an integer, leave it in simplest radical form.
9. 10. 11.
15. Architecture An escalator lifts people to the second floor, 25 ft above the first floor. The escalator rises at a 30° angle. How far does a person travel from the bottom to the top of the escalator? 43 ft 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?
Algebra Find the value of each variable. Leave your answer in simplest radical form.
17. 18. 19. b^1060 30
a
c d
b
60 45
4 (^3) a
c d
b 45 30
7 ^2 a
c d
BB Apply Your Skills x x^22
x
y
x
y 60 30
x^2 ^3 12
y
x
y^10
60
x y 60 30 2 3
(^40) x
y
Examples 4 and 5 x x^22
(page 427)
y
x
45 ^8
y
x
Examples 2 and 3 (page 426)
y 60
45
y (^) x
x
y
45
Example 1 (page 425)
Practice and Problem Solving
EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice.
AA Practice by Example
428 Chapter 8 Right Triangles and Trigonometry
x ≠ 8; y ≠ 8 " 2
x ≠ 15; y ≠ 15 4 " 2 "^10
x ≠ 20; y ≠ 20 " 3 x^ ≠^ "^3 ;^ y^ ≠^^3 x ≠ 5; y ≠ 5 " 3
x ≠ " 2 ; y ≠ (^2) y ≠ 60 " 2
Exercise 7
17. a ≠ 7; b ≠ 14; c ≠ 7; d ≠ 7 18. a ≠ 6; b ≠ 6 ; c ≠ 2 ; d ≠ 6 19. a ≠ 10 ; b ≠ 5 ; c ≠ 15; d ≠ 5
GO
for Help
x ≠ 24; y ≠ 12 " 3 x^ ≠^ 4;^ y^ ≠^^2 x^ ≠^ 9;^ y^ ≠^^18
25 ft
424 ft (^) 17–19. See above left.
25.5 ft
GPS Guided Problem Solving
Enrichment
Reteaching
Adapted Practice Name Class Date Practice 8-2 Similar Polygons Are the polygons similar? If they are, write a similarity statement, and givethe similarity ratio. If they are not, explain.
**1. 2. 3.
- 6.**
LMNO M HIJK. Complete the proportions and congruence statements.
10.^ 7.^ M^ =^? 11.8.^ K^ =^?^ 12.9.^ N^ =^? Algebra **The polygons are similar. Find the values of the variables.
- 16.**
k WXZ M k DFG. Use the diagram to find the following.
17. 18. the similarity ratio of m Z 19. (^) DGWXZ and DFG 20. GF 21. m G 22. m D 23. WZ
M
J (^) K x 15 m L 6 m9 m R SP Q YZ W E
X
F
G 1.5 cm H 4 cm 10 cm
x
S R O N
P Q 5 in. L M 3 in. A B E F 8 in. x
C 5 ft G 3.3 ft 6 ft x
MNIJJK? HK? LMHI MNIJ HK?
4
4 9 9 4 4
X YK 6 L
W ZN^^6 M
16 (^12 20 ) 21
B K CN M
8 A 8
8 8 8
8 8 8 60 60
120 R^120
S U
J K T M L
3
(^3 ) 5 5 5 B
A Y C
X Z 5
4 4
Q R
M N
10 12 20 — 3 20 — (^3) S 3 T 5 6 14
7
A B C
Y
X Z
Z
G
W^374 X D 6 F 3
M J^ K I H
N L O
Practice
L L L L L
- Practice
Assignment Guide
A B 1-8, 21, 22, 26
A B 9-20, 23-
C Challenge 27-
Test Prep 29-
Mixed Review 33-
Homework Quick Check
To check students’ understanding of key skills and concepts, go over Exercises 4, 12, 18, 24, 25.
Exercise 7 Ask a volunteer to
bring chopsticks to class and demonstrate how to use them.
Exercises 17–19 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.
Exercises 20–22 Each of these
exercises requires constructing an altitude to form a rectangle.
24. Answers may vary.
Sample: A ramp up to a
door is 12 ft long. It has
an incline of 30°. How
high off the ground is the
door? sol.: 6 ft
Lesson Quiz
Use k ABC for Exercises 1–3.
1. If m & A = 45, find AC and AB. AC ≠ 18; AB ≠ 18 2. If m & A = 30, find AC and AB. AC ≠ 18 ; AB ≠ 36 3. If m & A = 60, find AC and AB. AC ≠ 6 ; AB ≠ 12 4. Find the side length of a 45°-45°-90° triangle with a 4-cm hypotenuse. 2 N 2.8 cm 5. Two 12-mm sides of a triangle form a 120° angle. Find the length of the third side. 12 N 20.8 mm
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
Resources
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
A
C (^) 18 B
PowerPoint
- Assess & Reteach
Lesson 8-2 Special Right Triangles 429
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. 24. Open-Ended Write a real-life problem that you can solve using a 30 8 -60 8 -90 8 triangle with a 12 ft hypotenuse. Show your solution. 25. Farming A conveyor belt carries bales of hay from the ground to the barn loft 24 ft above the ground. The belt makes a 60 8 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 26. House Repair After heavy winds damaged a farmhouse, workers placed a 6-m brace against its side at a 45 8 angle. Then, at the same spot on the ground, they placed a second, longer brace to make a 30 8 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°-60°-90° triangle given a segment that is a. the shorter leg. b. the hypotenuse. c. the longer leg. See back of book. 29. What is the length of a diagonal of a square with sides of length 4? D A. 2 B. C. 2 D. 4 30. The longer leg of a 30°-60°-90° triangle is 6. What is the length of the hypotenuse? H A. 2 B. 3 C. 4 D. 12 31. The hypotenuse of a 30°-60°-90° triangle is 30. What is the length of one of its legs? D A. 3! 10 B. 10! 3 C. 15! 2 D. 15
Multiple Choice
d
d
CC Challenge
6 m
30 ^ 60
a
b 45
b
a 45
b
a
Exercise 25
a ≠ 4; b ≠ 4 a^ ≠^ 3;^ b^ ≠^^7 a^ ≠^ 14;^ b^ ≠^^6 "^2
23. Rika; Sandra marked the shorter leg as opposite the 60 8 angle.
GPS
GO nline Homework Help Visit: PHSchool.com Web Code: aue-
27a. units b. 2 units c. s " 3 units
Test Prep
lesson quiz, PHSchool.com, Web Code: aua-
See margin.
