Geometry Unit 8 Right Triangles and Trigonometry

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Date Name of Lesson Days Suggested

8.1 Geometric Mean 1 day

8.2 Pythagorean Theorem and It’s Converse 2 days

8.3 Special Right Triangles 1 day

Mid Chapter Quiz 1 day

8.4 Trigonometry 2 days

8.5 Angles of Elevation and Depression 1 day

Practice Test 1 day

Test 1 day

Geometry Unit 8 Right Triangles and Trigonometry

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8.1 Geometry Mean

Example 1: Geometric Mean Find the geometric mean between 2 and 50.

Guided Practice 1: Geometric Mean Find the geometric mean between 3 and 12.

Example 2: Identifying Similar Right Triangles Write a similarity statement identifying the three similar triangles in the figure

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Guided Practice 2: Identifying Similar Right Triangles Write a similarity statement identifying the three similar triangles in the figure

Example 3: Using the Geometric Mean with Right Triangles Find the values of c, d, and e.

Guided Practice 3: Using the Geometric Mean with Right Triangles Find the values of e to the nearest tenth

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Example 4: Indirect Measurement KITES Ms. Alspach is making a kite for her son. She has to arrange two support rods so that they are perpendicular. The shorter rod is 27 inches long. If she has to place the short rod 7.25 inches from one end of the long rod in order to form two right angles with the kite fabric, what is the length of the long rod?

Guided Practice 4: Indirect Measurement AIRPLANES A jetliner has a wingspan, BD, of 211 feet. The segment drawn from the front of the plane to the tail, segment AC intersects segment BD at point E. If AE is 163 feet, what is the length of the aircraft to the nearest tenth of a foot?

Geometry Unit 8 Right Triangles and Trigonometry

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8.2 The Pythagorean Theorem and Its Converse

Example 1A: Find x

Guided Practice 1: Find x

Guided Practice 1A:

Guided Practice 1B

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Example 2: Using Pythagorean Triples Use a Pythagorean Triple to find x. Explain your reasoning.

Guided Practice 2: Using Pythagorean Triples Use a Pythagorean Triple to find x. Explain your reasoning.

Example 3: Use the Pythagorean Theorem 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?

Guided Practice 3: Use Pythagorean Theorem 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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Example 4A: Classify Triangles 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. Guided Practice 4A: Classify Triangles Determine whether the set of numbers 7, 8, and 14 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.

Guided Practice 4B: Classify Triangles 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. Guided Practice 4B: Classify Triangles Determine whether the set of numbers 14, 18, and 33 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.

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8.3 Exploring Special Right Triangles (45-45-90) Given the isosceles right triangle to the right… 1. What is the measure of each angle? Explain 2a. If the length of each leg is 1 unit, find the length of the hypotenuse. Leave your answer in simplified rational form. 2b. What are the side-length ratios of leg : leg : hypotenuse? 3. If the length of each leg is 2 unit, find the length of the hypotenuse. Leave your answer in simplified rational form.

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8.3 Special Right Triangles Part 1 (45-45-90)

Example 1A: Find x

Example 1B: Find x

Guided Practice 1A: Find x Guided Practice 1B: Find x Example 2: Find a Guided Practice 2 Find b

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8.3 Exploring Special Right Triangles (30-60-90) Given the equilateral triangle to the right whose sides are 2 units long… 1. What is the measure of each acute angle? 2. Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along the crease. Label point D where the new segment intersects BC. What is special about AD? What is the length of BD (label it in the diagram)? Explain your reasoning. 3. What does AD do to angle A? 4. Examine triangle ABD. What is the measure of angle BAD and BDA (Label it in the diagram above)? 5. Find the length of AD. Leave your answer in simplified radical notation.

Geometry Unit 8 Right Triangles and Trigonometry

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8.3 Special Right Triangles Part 2 (30-60-90)

Example 1: Find x and y

Example 2: Find x and y Example 3: Using Special Right Triangles A quilt has the design shown in the figure, in which a square is divided into 8 isosceles right triangles. If the length of one side of the square is 3 inches, what are the dimensions of each triangle?

Guided Practice 1: BA and BC

Guided Practice 2: Find x and y Guided Practice 3: Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends.

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8.4 Trigonometry

Example 1: Sine, Cosine, and Tangent Express the following functions as fractions and a as a decimal to the nearest hundredth. sin L cos L tan L Guided Practice 1: Sine, Cosine, and Tangent sin A cos A tan A

sin N cos N tan N sin B cos B tan B

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Example 2: Estimate Using Trigonometry 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?

Guided Practice 2: Estimate Using Trigonometry 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?

Example 3: Inverse Trig Ratios Use a calculator to find the measure of ∠P to the nearest tenth.

Guided Practice 3: Inverse Trig Ratios Use a calculator to find the measure of ∠D to the nearest tenth.

Example 4: Solve the Right Triangle Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.

Guided Practice 4: Solve the Right Triangle Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.

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8.5 Symmetry

Example 1: Angle of Elevation 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 base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat’s head is 27°?

Guided Practice 1: Angle of Elevation 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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Example 2: Angle of Depression Maria is at the top of a cliff and sees a seal in the water. If the cliff 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?

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Guided Practice 2: Angle of Depression Luisa is in a hot air balloon 30 feet above the ground. She sees the landing spot at an angle of depression of 34° . What is the horizontal distance between the hot air balloon and the landing spot to the nearest foot? Example 3: Using Two Angles 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. Guided Practice 3: Using Two Angles Madison looks out her second-floor window, which is 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.

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