Download Geometry: Isosceles Right Triangles and the Pythagorean Theorem and more Lecture notes Trigonometry in PDF only on Docsity!
Washington Monument (p. 495)
Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems
arising in everyday life, society, and the workplace.
9.1 The Pythagorean Theorem 9.2 Special Right Triangles 9.3 Similar Right Triangles 9.4 The Tangent Ratio 9.5 The Sine and Cosine Ratios 9.6 Solving Right Triangles 9.7 Law of Sines and Law of Cosines
Right Triangles and
Trigonometry
Leaning Tower of Pisa (p. 518)
Fire Escape (p. 473)
Rock Wall (p. 485)
Skiing (p. 501)
Leaning Tower of Pisa (p. 518)
Skiing (p 501)
Washington Monument (p 495)
FiFire EEscape (((p. 4747 )3)3)
Rock Wall (p 485)
SEE the Big Idea
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Using Properties of Radicals (A.11.A)
Example 1 Simplify √
— 128.
√
— 128 = √
— (^64) ⋅ 2 Factor using the greatest perfect square factor. = √
— (^64) ⋅√
— 2 Product Property of Radicals = 8 √
— 2 Simplify.
Example 2 Simplify
— √
— 5
— √
— 5
— √
— 5
⋅
— 5 — √
— 5
Multiply by
— 5 — √
— 5
— 5 — √
— 25
Product Property of Radicals
— —^5 5
Simplify.
Simplify the expression.
1. √
— 75 2. √
— 270 3. √
— 135
— √
— 7
— √
— 2
— √
— 6
Solving Proportions (7.4.D)
Example 3 Solve (^) — x 10
= (^) —^3 2
—^ x 10
Write the proportion.
x (^) ⋅ 2 = (^10) ⋅ 3 Cross Products Property 2 x = 30 Multiply.
—^2 x 2
Divide each side by 2.
x = 15 Simplify.
Solve the proportion.
7. (^) — x 12
8. (^) — x 3
9.^4 —
x
= —^7
10.^10 —
x
11. x —^ +^1 2
12. —^9
3 x − 15
= —^3
13. ABSTRACT REASONING The Product Property of Radicals allows you to simplify the square root of a product. Are you able to simplify the square root of a sum? of a difference? Explain.
Section 9.1 The Pythagorean Theorem 467
9.1 The Pythagorean Theorem
Essential QuestionEssential Question How can you prove the Pythagorean Theorem?
Proving the Pythagorean Theorem
without Words
Work with a partner. a. Draw and cut out a right triangle with legs a and b , and hypotenuse c.
b. Make three copies of your right triangle. Arrange all four triangles to form a large square, as shown.
c. Find the area of the large square in terms of a , b , and c by summing the areas of the triangles and the small square.
d. Copy the large square. Divide it into two smaller squares and two equally-sized rectangles, as shown.
e. Find the area of the large square in terms of a and b by summing the areas of the rectangles and the smaller squares.
f. Compare your answers to parts (c) and (e). Explain how this proves the Pythagorean Theorem.
Proving the Pythagorean Theorem
Work with a partner. a. Draw a right triangle with legs a and b , and hypotenuse c , as shown. Draw the altitude from C to AB —. Label the lengths, as shown.
b a
A D B
C
c
h
c − d d
b. Explain why △ ABC , △ ACD , and △ CBD are similar. c. Write a two-column proof using the similar triangles in part (b) to prove that a^2 + b^2 = c^2.
Communicate Your AnswerCommunicate Your Answer
3. How can you prove the Pythagorean Theorem? 4. Use the Internet or some other resource to find a way to prove the Pythagorean Theorem that is different from Explorations 1 and 2.
REASONING
To be profi cient in math, you need to know and fl exibly use different properties of operations and objects.
a
a
a
a
b
b
b
b
c
c c
c
a
a
a
a
b
b
b b
G.6.D G.9.B
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
468 Chapter 9 Right Triangles and Trigonometry
9.1 Lesson What You Will LearnWhat You Will Learn
Use the Pythagorean Theorem. Use the Converse of the Pythagorean Theorem. Classify triangles.
Using the Pythagorean Theorem One of the most famous theorems in mathematics is the Pythagorean Theorem, named for the ancient Greek mathematician Pythagoras. This theorem describes the relationship between the side lengths of a right triangle.
Using the Pythagorean Theorem
Find the value of x. Then tell whether the side lengths form a Pythagorean triple.
SOLUTION
c^2 = a^2 + b^2 Pythagorean Theorem x^2 = 52 + 122 Substitute. x^2 = 25 + 144 Multiply. x^2 = 169 Add. x = 13 Find the positive square root.
The value of x is 13. Because the side lengths 5, 12, and 13 are integers that satisfy the equation c^2 = a^2 + b^2 , they form a Pythagorean triple.
Pythagorean triple, p. 468 Previous right triangle legs of a right triangle hypotenuse
Core VocabularyCore Vocabullarry
TheoremTheorem
Theorem 9.1 Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Proof Explorations 1 and 2, p. 467; Ex. 39, p. 488
A Pythagorean triple is a set of three positive integers a , b , and c that satisfy the equation c^2 = a^2 + b^2.
CoreCore ConceptConcept
Common Pythagorean Triples and Some of Their Multiples
3 x , 4 x , 5 x
5 x , 12 x , 13 x
8 x , 15 x , 17 x
7 x , 24 x , 25 x The most common Pythagorean triples are in bold. The other triples are the result of multiplying each integer in a bold-faced triple by the same factor.
STUDY TIP
You may fi nd it helpful to memorize the basic Pythagorean triples, shown in bold , for standardized tests.
a
b
c
5
12
x
c^2 = a^2 + b^2
470 Chapter 9 Right Triangles and Trigonometry
Verifying Right Triangles
Tell whether each triangle is a right triangle. a. 8
113
7
b.
36
4 95 15
SOLUTION
Let c represent the length of the longest side of the triangle. Check to see whether the side lengths satisfy the equation c^2 = a^2 + b^2.
a. (√
— 113 )
2
113 = 113 ✓
The triangle is a right triangle.
b. (^4 √
— 95 )
2
(^42) ⋅(√
— 95 )
2
(^16) ⋅ 95 =
1520 ≠ 1521 ✗
The triangle is not a right triangle.
Monitoring ProgressMonitoring Progress (^) Help in English and Spanish at BigIdeasMath.com
Tell whether the triangle is a right triangle.
4. 9
3 34
15
26
(^2214)
SELECTING TOOLS
Use a calculator to determine that √
— 113 ≈ 10.630 is the length of the longest side in part (a).
Using the Converse of the Pythagorean Theorem The converse of the Pythagorean Theorem is also true. You can use it to determine whether a triangle with given side lengths is a right triangle.
TheoremTheorem
Theorem 9.2 Converse of the Pythagorean Theorem
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If c^2 = a^2 + b^2 , then △ ABC is a right triangle.
Proof Ex. 39, p. 474
a
B
C b A
c
Section 9.1 The Pythagorean Theorem 471
Classifying Triangles
Verify that segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle. Is the triangle acute , right , or obtuse?
SOLUTION
Step 1 Use the Triangle Inequality Theorem (Theorem 6.11) to verify that the segments form a triangle.
4.3 + 5.2 >
9.5 > 6.1 ✓ 10.4 > 5.2 ✓ 11.3 > 4.3 ✓
The segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle.
Step 2 Classify the triangle by comparing the square of the length of the longest side with the sum of the squares of the lengths of the other two sides. c^2 a^2 + b^2 Compare c^2 with a^2 + b^2. 6.1^2 4.3^2 + 5.2^2 Substitute. 37.21 18.49 + 27.04 Simplify. 37.21 < 45.53 c^2 is less than a^2 + b^2.
The segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form an acute triangle.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
6. Verify that segments with lengths of 3, 4, and 6 form a triangle. Is the triangle acute , right , or obtuse? 7. Verify that segments with lengths of 2.1, 2.8, and 3.5 form a triangle. Is the triangle acute , right , or obtuse?
REMEMBER
The Triangle Inequality Theorem (Theorem 6.11) on page 343 states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Classifying Triangles The Converse of the Pythagorean Theorem is used to determine whether a triangle is a right triangle. You can use the theorem below to determine whether a triangle is acute or obtuse.
TheoremTheorem
Theorem 9.3 Pythagorean Inequalities Theorem
For any △ ABC , where c is the length of the longest side, the following statements are true. If c^2 < a^2 + b^2 , then △ ABC is acute. If c^2 > a^2 + b^2 , then △ ABC is obtuse.
a (^) B
A
C
b c
a (^) B
A
C
b
c
Proof Exs. 42 and 43, p. 474
c^2 < a^2 + b^2 c^2 > a^2 + b^2
Section 9.1 The Pythagorean Theorem 473
13. MODELING WITH MATHEMATICS The fire escape forms a right triangle, as shown. Use the Pythagorean Theorem (Theorem 9.1) to approximate the distance between the two platforms. (See Example 3.)
16.7 ft x
8.9 ft
14. MODELING WITH MATHEMATICS The backboard of the basketball hoop forms a right triangle with the supporting rods, as shown. Use the Pythagorean Theorem (Theorem 9.1) to approximate the distance between the rods where they meet the backboard.
x 13.4 in.
9.8 in.
In Exercises 15 –20, tell whether the triangle is a right triangle. (See Example 4.)
15.
65
72
97
14 10
2 6
3 5
In Exercises 21–28, verify that the segment lengths form a triangle. Is the triangle acute , right , or obtuse****? (See Example 5.)
21. 10, 11, and 14 22. 6, 8, and 10 23. 12, 16, and 20 24. 15, 20, and 36 25. 5.3, 6.7, and 7.8 26. 4.1, 8.2, and 12. 27. 24, 30, and 6√
— 43 28. 10, 15, and 5√
— 13
29. MODELING WITH MATHEMATICS In baseball, the lengths of the paths between consecutive bases are 90 feet, and the paths form right angles. The player on first base tries to steal second base. How far does the ball need to travel from home plate to second base to get the player out? 30. REASONING You are making a canvas frame for a painting using stretcher bars. The rectangular painting will be 10 inches long and 8 inches wide. Using a ruler, how can you be certain that the corners of the frame are 90°?
In Exercises 31–34, find the area of the isosceles triangle.
31.
17 m^ 17 m
16 m
h
10 cm^ 10 cm
12 cm
h
50 m 50 m
28 m
h
23
5 1
26
80
89
39
20 ft 20 ft
32 ft
h
474 Chapter 9 Right Triangles and Trigonometry
35. ANALYZING RELATIONSHIPS Justify the Distance Formula using the Pythagorean Theorem (Thm. 9.1). 36. HOW DO YOU SEE IT? How do you know ∠ C is a right angle without using the Pythagorean Theorem (Theorem 9.1)?
8
10
6
A B
C
37. PROBLEM SOLVING You are making a kite and need to figure out how much binding to buy. You need the binding for the perimeter of the kite. The binding comes in packages of two yards. How many packages should you buy? 38. PROVING A THEOREM Use the Pythagorean Theorem (Theorem 9.1) to prove the Hypotenuse-Leg (HL) Congruence Theorem (Theorem 5.9). 39. PROVING A THEOREM Prove the Converse of the Pythagorean Theorem (Theorem 9.2). ( Hint : Draw △ ABC with side lengths a , b , and c , where c is the length of the longest side. Then draw a right triangle with side lengths a , b , and x , where x is the length of the hypotenuse. Compare lengths c and x .) 40. THOUGHT PROVOKING Consider two integers m and n , where m > n. Do the following expressions produce a Pythagorean triple? If yes, prove your answer. If no, give a counterexample. 2 mn , m^2 − n^2 , m^2 + n^2 41. MAKING AN ARGUMENT Your friend claims 72 and 75 cannot be part of a Pythagorean triple because 722 + 752 does not equal a positive integer squared. Is your friend correct? Explain your reasoning. 42. PROVING A THEOREM Copy and complete the proof of the Pythagorean Inequalities Theorem (Theorem 9.3) when c^2 < a^2 + b^2. Given In △ ABC , c^2 < a^2 + b^2 , where c is the length of the longest side. △ PQR has side lengths a , b , and x , where x is the length of the hypotenuse, and ∠ R is a right angle. Prove △ ABC is an acute triangle.
A
B C Q R
P
a
c b
a
x b
STATEMENTS REASONS
1. In △ ABC , c^2 < a^2 + b^2 , where c is the length of the longest side. △ PQR has side lengths a , b , and x , where x is the length of the hypotenuse, and ∠ R is a right angle.
1. _______________
2. a^2 + b^2 = x 2 2. _______________ 3. c^2 < x^2 3. _______________ 4. c < x (^) 4. Take the positive square root of each side. 5. m ∠ R = 90 ° 5. _______________ 6. m ∠ C < m ∠ R 6.^ Converse of the Hinge Theorem (Theorem 6.13) 7. m ∠ C < 90° 7. _______________ 8. ∠ C is an acute angle. 8.^ _______________ 9. △ ABC is an acute triangle. 9. _______________ 43. PROVING A THEOREM Prove the Pythagorean Inequalities Theorem (Theorem 9.3) when c^2 > a^2 + b^2. (Hint : Look back at Exercise 42.)
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Simplify the expression by rationalizing the denominator. (Skills Review Handbook)
44.
— √
— 2
— √
— 3
— √
— 2
— √
— 3
Reviewing what you learned in previous grades and lessons
ng 15 in.
20 in.
12 in. 12 in.
476 Chapter 9 Right Triangles and Trigonometry
9.2 Lesson What You Will LearnWhat You Will Learn
Find side lengths in special right triangles. Solve real-life problems involving special right triangles.
Finding Side Lengths in Special Right Triangles A 45°- 45°- 90° triangle is an isosceles right triangle that can be formed by cutting a square in half diagonally.
Finding Side Lengths in 45 ° - 45 ° - 90 ° Triangles
Find the value of x. Write your answer in simplest form. a.
x
8 45 °
b.
x x
5 2
SOLUTION
a. By the Triangle Sum Theorem (Theorem 5.1), the measure of the third angle must be 45°, so the triangle is a 45°- 45°- 90° triangle. hypotenuse = leg (^) ⋅√
— 2 45 °- 45°- 90° Triangle Theorem x = (^8) ⋅√
— 2 Substitute. x = 8 √
— 2 Simplify.
The value of x is 8√
—
b. By the Base Angles Theorem (Theorem 5.6) and the Corollary to the Triangle Sum Theorem (Corollary 5.1), the triangle is a 45°- 45°- 90° triangle. hypotenuse = leg (^) ⋅√
— 2 45 °- 45°- 90° Triangle Theorem 5 √
— 2 = x (^) ⋅√
— 2 Substitute. 5 √
— 2 — √
— 2
x √
— 2 — √
— 2
Divide each side by √
—
5 = x Simplify.
The value of x is 5.
REMEMBER
A radical with index 2 is in simplest form when no radicands have perfect squares as factors other than 1, no radicands contain fractions, and no radicals appear in the denominator of a fraction.
Previous isosceles triangle
Core VocabularyCore Vocabullarry
TheoremTheorem
Theorem 9.4 45 °- 45°- 90° Triangle Theorem
In a 45°- 45°- 90° triangle, the hypotenuse is √
— 2 times as long as each leg.
Proof Ex. 19, p. 480 hypotenuse^ =^ leg^ ⋅√
— 2
x
x 45 ° x^^2
45 °
Section 9.2 Special Right Triangles 477
Finding Side Lengths in a 30 ° - 60 ° - 90 ° Triangle
Find the values of x and y. Write your answer in simplest form.
SOLUTION
Step 1 Find the value of x. longer leg = shorter leg (^) ⋅√
— 3 30 °- 60°- 90° Triangle Theorem 9 = x (^) ⋅√
— 3 Substitute. 9 — √
— 3
= x Divide each side by √
—
— √
— 3
⋅
— 3 — √
— 3
= x Multiply by
— 3 — √
— 3
— 3 — 3 = x Multiply fractions.
3 √
— 3 = x Simplify.
The value of x is 3√
—
Step 2 Find the value of y. hypotenuse = shorter leg (^) ⋅ 2 30 °- 60°- 90° Triangle Theorem y = 3 √
— (^3) ⋅ 2 Substitute. y = 6 √
— 3 Simplify.
The value of y is 6√
—
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Find the value of the variable. Write your answer in simplest form.
1.
x
x
2 2
y
2 2
x
3 30 °
60 °
h 4 4
2 2
REMEMBER
Because the angle opposite 9 is larger than the angle opposite x , the leg with length 9 is longer than the leg with length x by the Triangle Larger Angle Theorem (Theorem 6.10).
TheoremTheorem
Theorem 9.5 30 °- 60°- 90° Triangle Theorem
In a 30°- 60°- 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √
— 3 times as long as the shorter leg.
Proof Ex. 21, p. 480
hypotenuse = shorter leg (^) ⋅ 2 longer leg = shorter leg (^) ⋅√
— 3
2 x x
x 3
30 °
60 °
9
x
y
30 °
60 °
Section 9.2 Special Right Triangles 479
9.2 Exercises Tutorial Help in English and Spanish at BigIdeasMath.com
In Exercises 3–6, find the value of x****. Write your answer in simplest form. (See Example 1.)
3.
x
7
45 °
x
5 2
5 2
x x
x
9 45 °
In Exercises 7–10, find the values of x and y. Write your answers in simplest form. (See Example 2.)
7.
9
x
y
30 °
x (^) y 60 °
3 3
x
y
60 ° (^24)
ERROR ANALYSIS In Exercises 11 and 12, describe and correct the error in finding the length of the hypotenuse.
11.
By the Triangle Sum Theorem (Theorem 5.1), the measure of the third angle must be 60 °. So, the triangle is a 30 ° - 60 ° - 90 ° triangle. hypotenuse = shorter leg (^) ⋅√
— 3 = 7 √
— 3 So, the length of the hypotenuse is 7 √
— 3 units.
✗
30 °
7
By the Triangle Sum Theorem (Theorem 5.1), the measure of the third angle must be 45 °. So, the triangle is a 45 ° - 45 ° - 90 ° triangle. hypotenuse = leg (^) ⋅ leg (^) ⋅√
— 2 = 5 √
— 2 So, the length of the hypotenuse is 5 √
— 2 units.
✗ 45 ° 5
5
In Exercises 13 and 14, sketch the figure that is described. Find the indicated length. Round decimal answers to the nearest tenth.
13. The side length of an equilateral triangle is 5 centimeters. Find the length of an altitude. 14. The perimeter of a square is 36 inches. Find the length of a diagonal.
In Exercises 15 and 16, find the area of the figure. Round decimal answers to the nearest tenth. (See Example 3.)
15.
8 ft
17. PROBLEM SOLVING Each half of the drawbridge is about 284 feet long. How high does the drawbridge rise when x is 30°? 45°? 60°? (See Example 4.)
284 ft
x
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. VOCABULARY Name two special right triangles by their angle measures. 2. WRITING Explain why the acute angles in an isosceles right triangle always measure 45°.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
y x
30 °
12 3
4 m
4 m
5 m
5 m
60 °
480 Chapter 9 Right Triangles and Trigonometry
18. MODELING WITH MATHEMATICS A nut is shaped like a regular hexagon with side lengths of 1 centimeter. Find the value of x. ( Hint : A regular hexagon can be divided into six congruent triangles.) 1 cm
x
19. PROVING A THEOREM Write a paragraph proof of the 45 °- 45°- 90° Triangle Theorem (Theorem 9.4). Given △ DEF is a 45°- 45°- 90° triangle. Prove The hypotenuse is √
— 2 times as long as each leg.
20. HOW DO YOU SEE IT? The diagram shows part of the Wheel of Theodorus.
1
1
1
(^1 )
1
1
2
3 4 5 6
7
a. Which triangles, if any, are 45°- 45°- 90° triangles? b. Which triangles, if any, are 30°- 60°- 90° triangles?
21. PROVING A THEOREM Write a paragraph proof of the 30°- 60°- 90° Triangle Theorem (Theorem 9.5). ( Hint : Construct △ JML congruent to △ JKL .) Given △ JKL is a 30°- 60°- 90° triangle. Prove The hypotenuse is twice as long as the shorter leg, and the longer leg is √
— 3 times as long as the shorter leg.
22. THOUGHT PROVOKING A special right triangle is a right triangle that has rational angle measures and each side length contains at most one square root. There are only three special right triangles. The diagram below is called the Ailles rectangle. Label the sides and angles in the diagram. Describe all three special right triangles.
60 °
2 2
23. WRITING Describe two ways to show that all isosceles right triangles are similar to each other. 24. MAKING AN ARGUMENT Each triangle in the diagram is a 45°- 45°- 90° triangle. At Stage 0, the legs of the triangle are each 1 unit long. Your brother claims the lengths of the legs of the triangles added are halved at each stage. So, the length of a leg of a triangle added in Stage 8 will be — 2561 unit. Is your brother correct? Explain your reasoning.
1
Stage 0
1
Stage 1 Stage 2
Stage 3 Stage 4
25. USING STRUCTURE △ TUV is a 30°- 60°- 90° triangle, where two vertices are U (3, −1) and V (−3, −1), UV^ — is the hypotenuse, and point T is in Quadrant I. Find the coordinates of T.
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Find the value of x****. (Section 8.1)
26. △ DEF ∼ △ LMN 27. △ ABC ∼ △ QRS
12 20
30
x
D F
E
N L
M
7
4 x
A C
B Q
R
S
Reviewing what you learned in previous grades and lessons
45 °
D
F E
45 °
x
x
60 ° 30 °
K
M
J L
482 Chapter 9 Right Triangles and Trigonometry
9.3 Lesson What You Will LearnWhat You Will Learn
Identify similar triangles. Solve real-life problems involving similar triangles. Use geometric means.
Identifying Similar Triangles When the altitude is drawn to the hypotenuse of a right triangle, the two smaller triangles are similar to the original triangle and to each other.
Identifying Similar Triangles
Identify the similar triangles in the diagram.
SOLUTION
Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation.
R
S
R U T
T
S
T U
△ TSU ∼ △ RTU ∼ △ RST
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Identify the similar triangles. 1.
S R
T
Q 2.^ F
G
E H
geometric mean, p. 484 Previous altitude of a triangle similar figures
Core VocabularyCore Vocabullarry
TheoremTheorem
Theorem 9.6 Right Triangle Similarity Theorem
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
△ CBD ∼ △ ABC , △ ACD ∼ △ ABC ,
and △ CBD ∼ △ ACD.
Proof Ex. 45, p. 488
A B
C
D
A B
C C
D D
R
U S
T
Section 9.3 Similar Right Triangles 483
Solving Real-Life Problems
Modeling with Mathematics
A roof has a cross section that is a right triangle. The diagram shows the approximate dimensions of this cross section. Find the height h of the roof.
Z W X
Y
5.5 m (^) 3.1 m h
6.3 m
SOLUTION
1. Understand the Problem You are given the side lengths of a right triangle. You need to find the height of the roof, which is the altitude drawn to the hypotenuse. 2. Make a Plan Identify any similar triangles. Then use the similar triangles to write a proportion involving the height and solve for h. 3. Solve the Problem Identify the similar triangles and sketch them.
X W W X
Z
Z
Y
Y
Y
5.5 m^ 5.5 m 3.1 m
3.1 m
h
h
6.3 m
△ XYW ∼ △ YZW ∼ △ XZY Because △ XYW ∼ △ XZY , you can write a proportion. YW — ZY
= XY —
XZ
Corresponding side lengths of similar triangles are proportional.
h —
Substitute.
h ≈ 2.7 Multiply each side by 5.5. The height of the roof is about 2.7 meters.
4. Look Back Because the height of the roof is a leg of right △ YZW and right △ XYW , it should be shorter than each of their hypotenuses. The lengths of the two hypotenuses are YZ = 5.5 and XY = 3.1. Because 2.7 < 3.1, the answer seems reasonable.
Monitoring ProgressMonitoring Progress (^) Help in English and Spanish at BigIdeasMath.com
Find the value of x****.
3.
G F
E H
4
5
x
3
L
J M
K
12
x^5
13
COMMON ERROR
Notice that if you tried to write a proportion using △ XYW and △ YZW , then there would be two unknowns, so you would not be able to solve for h.
