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GEOMETRY UNIT 3

WORKBOOK

CHAPTER 8

Right Triangles & Trigonometry

SPRING 2017

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Geomety Section 8.1 Notes: Geometric Mean

Practice Makes Perfect! Example 1: Write a similarity statement identifying the three similar triangles in the figure. Remember to re-draw the three triangles. a)

Follow the directions and then answer the questions!

1) How are the two smaller right triangles related to the large triangle? 2) Explain how you would show that the green triangle is similar to the red triangle. 3) Explain how you would show that the red triangle is similar to the blue triangle. 4) What is true if the triangles are similar? (What can you set-up to solve for a missing side?)

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Example 2: a) Write a similarity statement identifying the three similar triangles in the figure. Remember to re-draw the three triangles.

b) Now you know that the sides are _____________________. Fill in the missing blanks.

1) 𝑆𝑈𝑇𝑈

=𝑅𝑈

2) 𝑇𝑆

= 𝑇𝑆𝑅𝑆

3) 𝑈𝑅 = 𝑇𝑅𝑆𝑅

Geometric Mean: Example 3: a) Find the geometric mean between 2 and 50. b) Find the geometric mean between 3 and 12. By definition of similar polygons, you can write proportions comparing the side lengths of these triangles.

Notice that the circled relationships involve geometric means.

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Radical Review Simplify the following

a) √𝟕𝟐 b) √𝟖𝟎 c) √𝟒𝟓 d) √𝟓𝟐

Parachute Pete Story *Pete always parachutes from the RIGHT ANGLE of the LARGE triangle. *The path he travels is the GEOMETRIC MEAN (so it is used TWICE in the proportion). *Then he visits TWO cities: (these are the other two blanks in the proportion). If the path he traveled was the ALTITUDE path, then he visits the LEFT city or the RIGHT city. If the path he traveled was an OUTSIDE path, then he visits the CLOSE city or the FAR city.

Use the letter P for Pete.

Path 1: Path 2: Path 3:

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Example 4:

a) Find c, d, and e. b) Find e. Example 5: Solve for the missing side length.

a) b)

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x 6

x

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Geometry Name: _____________________________________ Section 8.1 Practice Worksheet Find the geometric mean between each pair of numbers. 1. 8 and 12 2. 3 and 15 3. 4

5 and 2

For numbers 4 and 5, write a similarity statement identifying the three similar triangles in the figure. Remember to re-draw all three triangles. 4. 5. For numbers 6 – 9, find x, y, and z. Leave answer in simplified radical form. 6. 7. 8. 9. 10. CIVIL An airport, a factory, and a shopping center are at the vertices of a right triangle formed by three highways. The airport and

factory are 6.0 miles apart. Their distances from the shopping center are 3.6 miles and 4.8 miles, respectively. A service road will be constructed from the shopping center to the highway that connects the airport and factory. What is the shortest possible length for the service road? Round to the nearest hundredth.

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11. SQUARES Wilma has a rectangle of dimensions ℓ by w. She would like to replace it with a square that has the same area. What is

the side length of the square with the same area as Wilma’s rectangle? 12. EQUALITY Gretchen computed the geometric mean of two numbers. One of the numbers was 7 and the geometric mean turned

out to be 7 as well. What was the other number? 13. VIEWING ANGLE A photographer wants to take a picture of a beach front. His camera has a viewing angle of 90° and he wants

to make sure two palm trees located at points A and B in the figure are just inside the edges of the photograph.

He walks out on a walkway that goes over the ocean to get the shot. If his camera has a viewing angle of 90°, at what distance down the walkway should he stop to take his photograph?

14. EXHIBITIONS A museum has a famous statue on display. The curator places the statue in the corner of a rectangular room and builds a 15-foot-long railing in front of the statue. Use the information below to find how close visitors will be able to get to the statue.

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Geometry Section 8.2 Notes: The Pythagorean Theorem and its Converse Label each side of the right triangles with leg, leg, and hypotenuse.

a) b)

Example 1: Find the value of x. a) b) Pythagorean Triples

Example 2: Use a Pythagorean Triple to find the value of x. a) b)

c) d)

x

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Pythagorean Theorem (your bff):

x

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Example 3: a) A 20-foot ladder is placed against a building to reach a window that is 16 feet above the ground. How many feet away from the building is the bottom of the ladder? b) A 10-foot ladder is placed against a building. The base of the ladder is 6 feet from the building. How high does the ladder reach on the building?

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A rug designer decided to make a rug consisting of three square pieces sew together at their corners (called a Tri-Square Rug). Al and Betty use these rugs and turn it into a game: Let a dart fall randomly on the tri-square rug. If it hits the largest square, Al wins. If it hits either of the two other squares, Betty wins. If the dart misses the rug, let another dart fall. There are six games shown here for you. First circle the rug that belongs to Al. Then figure out who will be the winner (if any), and show your work as to why you think so. (Hint: If Betty has more total area than Al, she will have a higher probability of winning. What do you think is true if Betty and Al have the same area? __________________________________) Game 1 Game 2 Game 3 Game 4 Game 5 Game 6

Tri-Square Rug Games

Winner: _____________

Why?

Winner: _____________

Why?

Winner: _____________

Why?

Winner: _____________

Why?

Winner: _____________

Why?

Winner: _____________

Why?

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8 24

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21 20

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Discussion Questions! Write down your answers to each of the following and check!

1) a) What happens to the areas of the squares when you have a fair game?

b) Each of the rugs on the front create a triangle in the middle. What is special about the triangles formed with the fair games? (classify by its angles – you may want to use your protractor).

c) What theorem do you think of when you have that type of triangle? (do you see how it is derived using the games??)

2) a) What happens to the areas of the squares when Betty wins?

b) Each of the rugs on the front create a triangle in the middle. What is special about the triangles formed when Betty wins? (classify by its angles – you may want to use your protractor).

3) a) What happens to the areas of the squares when Al wins?

b) Each of the rugs on the front create a triangle in the middle. What is special about the triangles formed when Al wins? (classify by its angles – you may want to use your protractor).

4) Now generalize what you found without using numbers. If we labeled the sides of the squares with a, b, and c (c being the longest side),

a) Betty’s total area can be represented as: ____________________________ b) Al’s total area can be represented as: ______________________________ c) Now fill in the table provided:

Al Wins Betty Wins Fair Game c2 ____ a2 + b2

c2 ____ a2 + b2

c2 ____ a2 + b2

Note: BE SURE TO SHOW YOUR WORK in your HW for full credit!

Example 4: a) Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. b) Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.

c

b

a

Type of ∆ formed

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Geometry Name: _____________________________________ Section 8.2 Practice Worksheet For numbers 1 – 3, find x. 1. 2. 3. For numbers 4 – 7, use a Pythagorean Triple or the Pythagorean Theorem to find x. 4. 5. 6. 7. For numbers 8 – 13, determine whether each set of numbers can be measure of the sides of a triangle. If so, classify the triangle as acute, obtuse, or right. Justify your answer. You must show mathematical work for full credit. 8. 10, 11, 20 9. 12, 14, 49 10. 5√2, 10, 11 11. 21.5, 24, 55.5 12. 30, 40, 50 13. 65, 72, 97 14. CONSTRUCTION The bottom end of a ramp at a warehouse is 10 feet from the base of the main dock and is 11 feet long. How

high is the dock?

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15. SIDEWALKS Construction workers are building a marble sidewalk around a park that is shaped like a right triangle. Each marble slab adds 2 feet to the length of the sidewalk. The workers find that exactly 1071 and 1840 slabs are required to make the sidewalks along the short sides of the park. How many slabs are required to make the sidewalk that runs along the long side of the park?

16. RIGHT ANGLES Clyde makes a triangle using three sticks of lengths 20 inches, 21 inches, and 28 inches. Is the triangle a right

triangle? Explain. 17. TETHERS To help support a flag pole, a 50-foot-long tether is tied to the pole at a point 40 feet above the ground. The tether is

pulled taut and tied to an anchor in the ground. How far away from the base of the pole is the anchor? 18. FLIGHT An airplane lands at an airport 60 miles east and 25 miles north of where it took off.

How far apart are the two airports?

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Geometry Section 8.3 Notes: Special Right Triangles ABCD is a square. Draw diagonal BD. Classify ∆BDC by its sides. Label each of the angle measures.

Suppose DC = 1 cm. What is the length of the base (side opposite 45° angle)? Altitude (opposite 45° angle)? Hypotenuse (opposite 90° angle)? Let’s look for a pattern, given each base length in the table.

Base (in cm) (Side opposite

45o angle)

Altitude (opposite 45o angle)

Hypotenuse (Side opposite

90o angle)

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w

Example 1: Find the measure of each hypotenuse. a) b) Example 2: a) b)

A B

C D

Summary:

Note: First label each angle measurement; then label the sides.

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∆DOR is equilateral. Draw an altitude from O and name it OY . Label the measure of every angle.

Suppose DR = 10 cm. What is the length of the base (side opposite the 30° angle)? Altitude (opposite 60° angle)? Hypotenuse (opposite 90° angle)? Let’s look for a pattern, given each base length in the table.

Example 3: Solve for x and y. a) b) c) Find BC. . Example 4: An equilateral triangle has side lengths of 16 m. Find the length of the altitude of the triangle.

Base (in cm) (Side opposite 30o

Angle)

Altitude (Side opposite 60o

Angle)

Hypotenuse (Side opposite 90o

Angle)

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D R

O

Summary:

Note: First label each angle measurement; then label the sides.

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Geometry Name: _____________________________________ Section 8.3 Practice Worksheet For numbers 1 – 6, find x. 1. 2. 3. 4. 5. 6. For numbers 7 – 10, find x and y. 7. 8. 9. 10.

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11. Determine the length of the leg of 45°–45°–90° triangle with a hypotenuse length of 38. 12. Find the length of the hypotenuse of a 45°–45°–90° triangle with a leg length of 77 centimeters. 13. An equilateral triangle has an altitude length of 33 feet. Determine the length of a side of the triangle. 14. BOTANICAL GARDENS One of the displays at a botanical garden is an herb garden planted

in the shape of a square. The square measures 6 yards on each side. Visitors can view the herbs from a diagonal pathway through the garden. How long is the pathway?

15. ORIGAMI A square piece of paper 150 millimeters on a side is folded in half along a diagonal. The result is a 45°-45°-90°

triangle. What is the length of the hypotenuse of this triangle? 16. ESCALATORS A 40-foot-long escalator rises from the first floor to the second floor of a shopping mall. The escalator makes a

30° angle with the horizontal. How high above the first floor is the second floor?

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When do I use Trigonometry??

Geometry Section 8.4 Notes: Trigonometry

Choice 1 Choice 2

Trigonometry Summary Sine Cosine Tangent

Example 1 a) Express sin L as a fraction. d) Express sin N as a fraction. b) Express cos L as a fraction. e) Express cos N as a fraction. c) Express tan L as a fraction. f) Express tan N as a fraction.

Make sure your calculator is in the “degrees” mode:

(click “MODE” and select degrees)

Labeling the sides: Choose an angle at which to “stand”. You CANNOT choose the right angle!

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Trigonometry Summary

Sine Cosine Tangent

Practice Find the value of each variable. Round decimals to the nearest tenth.

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g

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HI

G

Example 2: A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Example 3: The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch?

Inverse Trigonometry Summary Sine Cosine Tangent

Find the measure of ∠D to the nearest tenth.

Find the measure of ∠P to the nearest tenth.

Find the measure of ∠A to the nearest tenth.

Example 4: Solve each right triangle. Round side measures to the nearest hundredth and angle measures to the nearest tenth. a) b)

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Geometry Name: _____________________________________ Section 8.4 Practice Worksheet For numbers 1 and 2, find sin L, cos L, tan L, sin M, cos M, and tan M. Express each ratio as a fraction. 1. ℓ = 15, m = 36, n = 39 2. ℓ = 12, m = 12√3, n = 24 For numbers 3 – 5, find x. Round to the nearest hundredth. 3. 4. 5. For numbers 6 – 8, use a calculator to find the measure of ∠B to the nearest tenth. 6. 7. 8. 9. GEOGRAPHY Diego used a theodolite to map a region of land for his class in geomorphology. To determine the elevation of a

vertical rock formation, he measured the distance from the base of the formation to his position and the angle between the ground and the line of sight to the top of the formation. The distance was 43 meters and the angle was 36°. What is the height of the formation to the nearest meter?

10. RADIO TOWERS Kay is standing near a 200-foot-high radio tower. Use the information in the figure to determine how far Kay

is from the top of the tower. Express your answer as a trigonometric function.

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11. RAMPS A 60-foot ramp rises from the first floor to the second floor of a parking garage. The ramp makes a 15° angle with the

ground. How high above the first floor is the second floor? Express your answer as a trigonometric function. 12. TRIGONOMETRY Melinda and Walter were both solving the same trigonometry problem. However, after they finished their

computations, Melinda said the answer was 52 sin 27° and Walter said the answer was 52 cos 63°. Could they both be correct? Explain.

13. LINES Jasmine draws line m on a coordinate plane. What angle does m make with the x-axis? Round your answer to the nearest degree.

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Geometry Section 8.5 Notes: Angles of Elevation and Depression

Example 1: a) The angle of elevation from Kennedy to the top of a hill is 49o. If Kennedy is 400 feet from the base of the hill, how high is the hill to the nearest foot? b) Sarah is at the top of a cliff and sees a seal in the water. If the Sarah is 40 feet above the water and the angle of depression is 52°, what is the horizontal distance from the seal to the cliff, to the nearest foot? Example 2: a) Find the angle of elevation of the Sun when a 12.5-meter-tall telephone pole casts an 18-meter-long shadow. Round to the nearest tenth. b) A ski run is 1000 yards long with a vertical drop of 208 yards. Find the angle of depression from the top of the ski run to the bottom to the nearest tenth of a degree.

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Example 3: a) At the circus, a person in the audience at ground level watches the high-wire routine. A 5-foot-6-inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the acrobat, if the angle of elevation from the audience member’s line of sight to the top of the acrobat is 27°? b) Drayton sees a hot air balloon in the sky from his spot on the ground. The angle of elevation from Drayton to the balloon is 40°. He is 200 feet away from the balloon when it’s in the air. If Drayton is 6 feet tall, how far off the ground is the hot air balloon? Example 4: a) Vernon is on the top deck of a cruise ship and observes two dolphins following each other directly away from the ship in a straight line. Vernon’s position is 154 meters above sea level, and the angles of depression to the two dolphins are 35° and 36°. Find the distance between the two dolphins to the nearest meter.

b) Madison looks out her second-floor window from 15 feet above the ground. She observes two parked cars. One car is parked along the curb directly in front of her window and the other car is parked directly across the street from the first car. The angles of depression of Madison’s line of sight to the cars are 17° and 31°. Find the distance between the two cars to the nearest foot. c) At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the opposite end of the pool even with the pool’s edge. If the camera is angled so that its line of sight extends to the top of the diver’s head, what is the camera’s angle of elevation to the nearest degree?

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Geometry Name: _____________________________________ Section 8.5 Practice Worksheet For numbers 1 and 2, name the angle of depression or angle of elevation in each figure 1. 2. 3. AIRPLANES An airplane is flying at a height of 2 miles above the ground. The horizontal distance from the airport to the point

directly under the airplane is 7.46 miles. What is the angle of depression of the airplane to the airport? 4. CONSTRUCTION A roofer props a 40-foot ladder against a wall. If the angle of elevation from the bottom of the ladder to the

roof is 55°, how far is the ladder from the base of the wall? Round your answer to the nearest tenth. 5. TOWN ORDINANCES The town of Belmont restricts the height of flagpoles to 25 feet

on any property. Lindsay wants to determine whether her school is in compliance with the regulation. Her eye level is 5.5 feet from the ground and she stands 36 feet from the flagpole. If the angle of elevation is about 25°, what is the height of the flagpole to the nearest tenth?

6. GEOGRAPHY Stephan is standing on the ground by a mesa in the Painted

Desert. Stephan is 1.8 meters tall and sights the top of the mesa at 29°. Stephan steps back 100 meters and sights the top at 25°. How tall is the mesa?

7. INDIRECT MEASUREMENT Mr. Dominguez is standing on a 40-foot ocean

bluff near his home. He can see his two dogs on the beach below. If his line of sight is 6 feet above the ground and the angles of depression to his dogs are 34° and 48°, how far apart are the dogs to the nearest foot?

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8. LIGHTHOUSES Sailors on a ship at sea spot the light from a lighthouse. The light of the lighthouse is 30 meters above sea level and the sailors are 63 meters away from the light of the lighthouse. What is the angle of elevation? Round to the nearest tenth. 9. RESCUE A hiker dropped his backpack over one side of a canyon onto a ledge below. Because of the shape of the cliff, he could

not see exactly where it landed. From the other side, the park ranger reports that the angle of depression to the backpack is 32°. If the width of the canyon is 115 feet, how far down did the backpack fall? Round your answer to the nearest tenth.

10. AIRPLANES The angle of elevation to an airplane viewed from the control tower at an airport is 7°. The tower is 200 feet high

and the pilot reports that the altitude is 5200 feet. How far away from the control tower is the airplane? Round your answer to the nearest foot.

11. PEAK TRAM The Peak Tram in Hong Kong connects two terminals, one at the base of a mountain, and the other at the summit.

The angle of elevation of the upper terminal from the lower terminal is about 15.5°. The distance between the two terminals is about 1365 meters. About how much higher above sea level is the upper terminal compared to the lower terminal? Round your answer to the nearest meter.