## Right Triangles And Trigonometry Cummulative Assessment

Question 1.The size of a laptop screen is measured by the length of its diagonal. you Want to purchase a laptop with the largest screen possible. Which laptop should you buy?

d = 9² + 12² = 15 d = 11.25² + 20² = 22.94 d = 12² + 6.75² = 13.76 d = 8² + 6² = 10

Question 2.In PQR and SQT, S is between P and Q, T is between R and Q, and \ What must be true about \ and \? Select all that apply.\ \ \ || \ ST = PR ST = \PRAnswer:

In the diagram, JKL ~ QRS. Choose the symbol that makes each statement true.< = > sin J ___________ sin Q sin L ___________ cos J cos L ___________ tan Qcos S ___________ cos J cos J ___________ sin S tan J ___________ tan Qtan L ___________ tan Q tan S ___________ cos Q sin Q ___________ cos LAnswer:sin J = sin Q sin L = cos J cos L = tan Qcos S > cos J cos J > sin S tan J = tan Qtan L < tan Q tan S > cos Q sin Q = cos L

Question 4.A surveyor makes the measurements shown. What is the width of the river.

Answer:tan 34 = \AB = 56.28

Create as many true equations as possible.

____________ = ______________

sin X cos X tan x \ \

Sin Z cos Z tan Z \ \

Answer:sin X = \ = cos Zcos X = \ = sin Ztan x = \tan Z = \

Question 6.Prove that quadrilateral DEFG is a kite.Given \, \ \Prove \, \Answer:

Question 7.What are the coordinates of the vertices of the image of QRS after the composition of transformations show? Q , R’, S’ Q’, R , S Q’, R , S Q , R’, S’Answer:

Answer:343² = h² + 722²h = 635.3

## The Sine And Cosine Ratios

**Exploration 1**

Work with a partner: Use dynamic geometry software.

a. Construct ABC, as shown. Construct segments perpendicular to \ to form right triangles that share vertex A arid are similar to ABC with vertices, as shown.Answer:

b. Calculate each given ratio to complete the table for the decimal values of sin A and cos A for each right triangle. What can you conclude?Answer:

Communicate Your Answer

Question 2.How is a right triangle used to find the sine and cosine of an acute angle? Is there a unique right triangle that must be used?Answer:

Question 3.In Exploration 1, what is the relationship between A and B in terms of their measures? Find sin B and cos B. How are these two values related to sin A and cos A? Explain why these relationships exist.**LOOKING FOR STRUCTURE**To be proficient in math, you need to look closely to discern a pattern or structure.Answer:

## Exercise 94 The Tangent Ratio

Vocabulary and Core Concept Check

Question 1.The tangent ratio compares the length of _________ to the length of ___________ .Answer:

tan 30° = \

Question 15.**MODELING WITH MATHEMATICS**A surveyor is standing 118 Feet from the base of the Washington Monument. The surveyor measures the angle of elevation from the ground to the top of the monument to be 78°. Find the height h of the Washington Monument to the nearest foot.Answer:

Question 16.**MODELING WITH MATHEMATICS**Scientists can measure the depths of craters on the moon h looking at photos of shadows. The length of the shadow cast by the edge of a crater is 500 meters. The angle of elevation of the rays of the Sun is 55°. Estimate the depth d of the crater.

Answer:tan 55 = \1.428 = \The depth of the crater is 714 m

Question 17.**USING STRUCTURE**Find the tangent of the smaller acute angle in a right triangle with side lengths 5, 12, and 13.Answer:

Question 18.**USING STRUCTURE**Find the tangent 0f the larger acute angle in a right triangle with side lengths 3, 4, and 5.

Answer:tan x = \

Question 19.**REASONING**How does the tangent of an acute angle in a right triangle change as the angle measure increases? Justify your answer.Answer:

Question 20.**CRITICAL THINKING**For what angle measure is the tangent of an acute angle in a right triangle equal to 1? greater than 1? less than 1? Justify your answer.

Answer:tan A = \ = \tan B = \ = \

Answer:

Answer:

Find the value of x.

Question 27.

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## Law Of Sines And Law Of Cosines

Find the area of ABC with the given side lengths and included angle.

Question 38.m B = 124°, a = 9, c = 11

Answer:Area = \ ac sin B= \ sin 124= 40.59

m A = 68°, b = 13, c = 7

Answer:Area = \ bc sin A= \ sin 68= 41.86

m C = 79°, a = 25 b = 17

Answer:Area = \ ab sin C= \ sin 79= 208.25

Solve ABC. Round decimal answers to the nearest tenth.

Question 41.m A = 112°, a = 9, b = 4

Answer:B = 24, C = 44, c = 6.76

Explanation:\ = \\ = \sin B = 0.408

sin 30 = \f = 4.6cos 30 = \h = 9.2

In QRS, m R = 57°, q = 9, and s = 5. Find the area of QRS.

Answer:Area = \ qs sin R= \ sin 57 = 18.675

Question 18.You are given the measures of both acute angles of a right triangle. Can you determine the side lengths? Explain.

Answer:No.

Question 19.You are at a parade looking up at a large balloon floating directly above the street. You are 60 feet from a point on the street directly beneath the balloon. To see the top of the balloon, you look up at an angle of 53°. To see the bottom of the balloon, you look up at an angle of 29°. Estimate the height h of the balloon.

Answer:

Question 20.You warn to take a picture of a statue on Easter Island, called a moai. The moai is about 13 feet tall. Your camera is on a tripod that is 5 feet tall. The vertical viewing angle of your camera is set at 90°. How far from the moai should you stand so that the entire height of the moai is perfectly framed in the photo?

Answer:

## Lesson 92 Special Right Triangles

**Monitoring Progress**

Find the value of the variable. Write your answer in simplest form.

Question 1.

longer leg = shorter leg 3h = 23

Question 5.The logo on a recycling bin resembles an equilateral triangle with side lengths of 6 centimeters. Approximate the area of the logo.

Answer:Area is \

Explanation:Area = \ a²= \²= \

Question 6.The body of a dump truck is raised to empty a load of sand. How high is the 14-foot-long body from the frame when it is tipped upward by a 60° angle?

Answer:28/3 ft high is the 14-foot-long body from the frame when it is tipped upward by a 60° angle.

Explanation:Height of body at 90 degrees = 14 ftHeight of body at 1 degree = 14/90Height of body at 60 degrees = 14 x 60/90= 14 x 2/3

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## Chapter 6 Test Form A Geometry Answers

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## Exercise 96 Solving Right Triangles

Question 1.To solve a right triangle means to find the measures of all its ________ and _______ .Answer:

Question 2.**WRITING**Explain when you can use a trigonometric ratio to find a side length of a right triangle and when you can use the Pythagorean Theorem .Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3 6. determine which of the two acute angles has the given trigonometric ratio.

Question 3.The cosine of the angle is \Answer:

The sine of the angle is \

Answer:Sin = \sin A = \The acute angle that has a sine of the angle is \ is A.

Question 5.The sine of the angle is 0.95.Answer:

The tangent of the angle is 1.5.

Answer:

Describe and correct the error in using an inverse trigonometric ratio.Answer:

Question 20.**PROBLEM SOLVING**In order to unload clay easily. the body of a dump truck must be elevated to at least 45° The body of a dump truck that is 14 feet long has been raised 8 feet. Will the clay pour out easily? Explain your reasoning.

Answer:Angle of elevation: sin x = \x = inverse sine of \ = 34.9The clay will not pour out easily.

Question 21.**PROBLEM SOLVING**You are standing on a footbridge that is 12 feet above a lake. You look down and see a duck in the water. The duck is 7 feet away from the footbridge. What is the angle of elevation from the duck to youAnswer:

Question 22.**HOW DO YOU SEE IT?**Write three expressions that can be used to approximate the measure of A. Which expression would you choose? Explain your choice.

Answer:

\Answer:

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## Exercise 95 The Sine And Cosine Ratios

Vocabulary and Core Concept Check

Question 1.The sine raio compares the length of ______________ to the length of _____________Answer:

**WHICH ONE DOESNT BELONG?**Which ratio does not belong with the other three? Explain your reasoning.sin B

sin B = \

cos C

cos C = \

tan B

Which ratios arc equal to \ Select allsin L

sin L = \ = \

cos L

cos L = \ = \

sin J

sin J = \ = \

cos J

cos J = \ = \

Question 25.Describe and correct the error in finding sin A.Answer:

Question 26.**WRITING**Explain how to tell which side of a right triangle is adjacent to an angle and which side is the hypotenuse.Answer:

Question 27.**MODELING WITH MATHEMATICS**The top of the slide is 12 feet from the ground and has an angle of depression of 53°. What is the length of the slide?Answer:

Find the horizontal distance x the escalator covers.

Answer:cos 41 = \0.754 = \

Question 29.**PROBLEM SOLVING**You are flying a kite with 20 feet of string extended. The angle of elevation from the spool of string to the kite is 67°.a. Draw and label a diagram that represents the situation.b. How far off the ground is the kite if you hold the spool 5 feet off the ground? Describe how the height where you hold the spool affects the height of the kite.Answer:

Answer:sin 1 = \0.017 = \h = 425 ft425 ft the plane rise to pass over the tower

b. PIanes Caillot come closer than 1000 feet vertically to any object. At what altitude must the plane fly in order to pass over the tower?

Answer:

Answer:sin = \cos = \

Answer:tan 34 = \.674 = \x = 2696

## Lesson 94 The Tangent Ratio

**Monitoring progress**

Find tan J and tan K. Write each answer as a fraction and as a decimal rounded to four places.

Question 1.

tan K = \ = \= \ = \ = 1.33tan J = \ = \ = \ = 0.75

Question 2.

tan K = \ = \tan J = \ = \

Find the value of x. Round your answer to the nearest tenth.

Question 3.

Tan 61 = \1.804 = \

tan 56 = \1.482 = \

Question 5.**WHAT IF?**In Example 3, the length of the shorter leg is 5 instead of 1. Show that the tangent of 60° is still equal to 3.

Answer:longer leg = shorter leg 3= 53tan 60 = \= 3

Question 6.You are measuring the height of a lamppost. You stand 40 inches from the base of the lamppost. You measure the angle ot elevation from the ground to the top of the lamppost to be 70°. Find the height h of the lamppost to the nearest inch.

Answer:tan 70 = \2.7474 = \

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## Lesson 95 The Sine And Cosine Ratios

**Monitoring Progress**

Question 1.Find sin D, sin F, cos D, and cos F. Write each answer as a fraction and as a decimal rounded to four places.Answer:sin D = \sin F = \cos D = \cos F = \

Explanation:sin D = \ = \sin F = \ = \cos D = \ = \cos F = \ = \

Question 2.Write cos 23° in terms of sine.

Answer:cos 23° = sin = sinSo, cos 23° = sin 67°

Question 3.Find the values of u and t using sine and cosine. Round your answers to the nearest tenth.Answer:

Find the sine and cosine of a 60° angle.

Answer:sin 60° = \cos 60° = \

Question 5.**WHAT IF?**In Example 6, the angle of depression is 28°. Find the distance x you ski down the mountain to the nearest foot.

Answer:sin 28° = \x = \

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## Exercise 92 Special Right Triangles

Vocabulary and Core Concept Check

Question 1.Name two special right triangles by their angle measures.Answer:

Answer:hypotenuse = leg 2 = 5 . 2 = 10

In Exercises 13 and 14. sketch the figure that is described. Find the indicated length. Round decimal answers to the nearest tenth.

Question 13.The side length of an equilateral triangle is 5 centimeters. Find the length of an altitude.Answer:

The perimeter of a square is 36 inches. Find the length of a diagonal.

Answer:The length of a diagonal is 92

Explanation:Side of the square = 36/4 = 9square diagonal = 2a = 2 = 92

In Exercises 15 and 16, find the area of the figure. Round decimal answers to the nearest tenth.

Question 15.

Area of the parallelogram = 5) = 40 sq m

Question 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°?Answer:

Question 18.**MODELING WITH MATHEMATICS**A nut is shaped like a regular hexagon with side lengths of 1 centimeter. Find the value of x.

Write a paragraph proof of the 30° 60° 90° Triangle Theorem .Given JKL is a 30° 60° 9o° 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.Answer:

Answer:

Describe two ways to show that all isosceles right triangles are similar to each other.Answer:

Find the Value of x.

Question 26.

**Exploration 1**

Writing a Conjecture

c. Use the proportion you wrote in part to find CD.Answer:

Answer:

Question 31.

Answer:

## Lesson 96 Solving Right Triangles

Determine which of the two acute angles has the given trigonometric ratio.

Question 1.The sine of the angle is \.

Answer:Sin E = \mE = sin-1) = 67.3°

Question 2.The tangent of the angle is \

Answer:tan F = \mF = tan-1) = 22.6°

Let G, H, and K be acute angles. Use a calculator to approximate the measures of G, H, and K to the nearest tenth of a degree.

Question 3.

sin G = \G = 56.09sin H = \H = 33.3

Solve the right triangle. Round decimal answers to the nearest tenth.

Answer:XY = 13.82, YZ = 6.69, Y = 37.5

Explanation:cos 52 = \0.615 = \sin 52 = \0.788 = \sin Y = \Y = 37.5

Question 9.**WHAT IF?**In Example 5, suppose another raked stage is 20 feet long from front to back with a total rise of 2 feet. Is the raked stage within your desired range?

Answer:x = inverse sine of \x = 5.7°

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