Geometry B

Ms. Hanning

Name: ______________________________________________ Hour: _________

Test Date: ____________________ Grade: _______/70 participation points

Chapter 7 Note Packet

Right Triangles

and

Trigonometry

To receive full credit on your notes, all notes must be completed and turned in on test day.

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3

Goals

Find side lengths in right triangles.

One of the most famous theorems in mathematics is the Pythagorean Theorem, named for the

ancient Greek mathematician Pythagoras (around 500 B.C.). This theorem can be used to find

information about the lengths of the sides of a right triangle.

Find the length of the hypotenuse of the right triangle.

A 16 foot ladder rests against the side of the house, and the base of the ladder is 4 feet

away. Approximately how high above the ground is the top of the ladder?

Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle.

Write your answer in simplest radical form.

1. 2.

3. A 5 foot board rests under a doorknob and the base of the board is 3.5 feet away from the bottom of the

door. Approximately how high above the ground is the doorknob?

Find the length of a hypotenuse Example 1

Find the length of a leg Example 2

Check Point

Section 7.1 – Apply the Pythagorean Theorem

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.

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Common Pythagorean Triples and some of Their Multiples

3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

6, 8, 10 10, 24, 26 16, 30, 34 14, 48, 50

9, 12, 15 15, 36, 39 24, 45, 51 21, 72, 75

30, 40, 50 50, 120, 130 80, 150, 170 70, 240, 250

3𝑥, 4𝑥, 5𝑥 5𝑥, 12𝑥, 13𝑥 8𝑥, 15𝑥, 17𝑥 7𝑥, 24𝑥, 25𝑥

The most common Pythagorean triples are in bold. The other triples are the result of

multiplying each integer in bold face triple by the same factor.

Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters.

Pythagorean Triples

A Pythagorean triple is set of three positive integers a, b, and c that satisfy the equation 𝑐2 = 𝑎2 + 𝑏2 .

Find the length of the hypotenuse of the right triangle.

Find the area of the triangle.

4. 5.

Find the unknown side length of the right triangle using the Pythagorean Theorem. Then use a

Pythagorean triple.

6. 7.

Find the length of a hypotenuse using two methods Example 4

Check Point

Find the area of an isosceles triangle Example 3

5

Goals: Find possible side lengths of a triangle.

Below is a drawing an obtuse scalene triangle. Notice that the longest side and the largest angle are opposite

each other and the shortest side and the smallest angle are opposite each other. These relationships are true for

all triangles are stated in the theorems below. These relationships can help you decide whether a particular

arrangement of side lengths and angle measures in a triangle may be possible.

You are constructing a stage prop that shows a large triangular mountain. The bottom edge of the mountain is

about 27 feet long, the left slope is about 24 feet long, and the right slope is about 20 feet long. You are told that

one of the angles is about 46° and one is about 59°. What is the angle measure of the peak of the mountain?

a) 46°

b) 59°

c) 75°

d) 85°

Pre-Requisite: Inequalities in Triangles

Standardized Test Practice Example 1

Theorems

Theorem 5.10

If one side of a triangle is longer than another side, then the angle

opposite the longer side is larger than the angle opposite the shorter side.

Theorem 5.11

If one angle of a triangle is larger than another angle, then the side

opposite the larger angle is longer than the side opposite the smaller angle.

Theorem 5.12

The sum of the lengths of any two sides of a triangle is greater than the

length of the third side.

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A triangle has one side length 12 and another of length 8. Describe the possible lengths of the third side.

List the sides and the angles in order from smallest to greatest.

1. 2.

Is it possible to construct a triangle with the given side lengths? If not, explain why not.

3. 6, 7, 11 4. 3, 6, 9

5. Describe the possible lengths of third side of the triangle with side lengths 3 meters and 4 meters.

Find the possible side lengths Example 2

Check Point

7

Goals: Use the Converse of the Pythagorean Theorem to determine if a triangle is a right triangle.

The converse of the Pythagorean Theorem is also true. You can use it to verify that a triangle with given side

lengths is a right triangle.

Tell whether the given triangle is a right triangle.

a) b)

Tell whether a triangle with the given side lengths is a right triangle.

1. 4, 4√3, and 8 2. 10, 11, and 14 3. 5, 6, and √61

a) b)

Section 7.2 – Use the Converse of the Pythagorean Theorem

Verify right triangles Example 1

Theorem 7.2: 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 𝑐2 = 𝑎2 + 𝑏2, then ∆𝐴𝐵𝐶 is a right triangle.

Check Point

Use Pythagorean Theorem on an Acute & Obtuse Triangle Example 2

4 in 7 in

6 in

5 mi

4 mi

8 mi 72 ? 42 + 62 82 ? 42 + 52

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Classifying Triangles

The Converse of the Pythagorean Theorem is used to verify that a given triangle is a right triangle. The

theorems summarized below are used to verify that a given triangle is acute or obtuse.

Can segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle? If so, would the triangle be acute,

right, or obtuse?

Tell whether the triangle is a right triangle. If it is not, classify it as acute, obtuse, or equiangular.

4. 5.

Decide whether the numbers can represent the side lengths of a triangle. If they can, classify the triangle

as acute, right, or obtuse.

6. 16, 30, and 34 7. 10, 12, and 30 8. 18, 34, and 45

Classify triangles Example 3

Step 1: Use the Triangle Inequality

Theorem to check that the segments an

make 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 shorter

sides.

Check Point

Methods for Classifying a Triangle by Angles Using its Side Lengths

Theorem 7.2 Theorem 7.3 Theorem 7.4

If 𝑐2 = 𝑎2 + 𝑏2, then,

𝑚∠𝐶 = 90° and ∆𝐴𝐵𝐶 is a

________________ triangle.

If 𝑐2 < 𝑎2 + 𝑏2, then,

𝑚∠𝐶 < 90° and ∆𝐴𝐵𝐶 is an

________________ triangle.

If 𝑐2 > 𝑎2 + 𝑏2, then,

𝑚∠𝐶 > 90° and ∆𝐴𝐵𝐶 is an

________________ triangle.

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Theorem 7.8: 𝟒𝟓°- 𝟒𝟓°- 𝟗𝟎° Triangle Theorem

In a 45°- 45°- 90° triangle, the hypotenuse is ______ times as long as each leg.

hypotenuse = leg ______

The extended ratio of the side lengths of a 45°- 45°- 90° triangle is ____ : ____ : ____

Pro

of

for

the

𝟒𝟓

°- 𝟒𝟓

°- 𝟗𝟎

° ∆

Goals: Use the relationships among the sides in special right triangles.

𝟒𝟓°- 𝟒𝟓°- 𝟗𝟎° Triangle.

Let’s see what the relationship is between the side lengths of a 45°- 45°- 90° Triangle. Draw a square below.

Find the length of the hypotenuse.

a) b)

Find the lengths of the legs in the triangle.

a) b)

Section 7.4 – Special Right Triangles

Check Point

Find hypotenuse length in a 45°- 45°- 90° triangle Example 1

Find leg lengths in a 45°- 45°- 90° triangle Example 2

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Pro

of

for

the

𝟑𝟎

°- 𝟔𝟎

°- 𝟗𝟎

° ∆

The body of a dump truck is raised to empty a load of sand. How high is the 14

foot body from the frame when it is tipped upward at a 45° angle?

Find the value of the variable.

1. 2. 3.

4. Approximate the length of the leg of a 45°- 45°- 90° triangle with a hypotenuse length of 6.

𝟑𝟎°- 𝟔𝟎°- 𝟗𝟎° Triangle.

Let’s see what the relationship is between the side lengths of a 30°- 60°- 90° Triangle. Draw an equilateral triangle

below.

Check Point

Find a height Example 3

Theorem 7.9: 𝟑𝟎°- 𝟔𝟎°- 𝟗𝟎° Triangle Theorem

In a 30°- 60°- 90° triangle, the hypotenuse is _________________ as long as the

shorter leg, and the longer leg is _______ times as long as the shorter leg.

hypotenuse = ______ shorter leg longer leg = shorter leg ______

The extended ratio of the side lengths of a 30°- 60°- 90° triangle is ____ : ____ : ____

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The logo on the recycling bin at the right resembles an equilateral triangle with side lengths

of 6 centimeters. What is the approximate height of the logo?

Approximate the values of x and y.

A car is turned off while the windshield wipers are moving. The 24 inch wipers

stop, making a 60° angle with the bottom of the windshield. How far from the

bottom of the windshield are the ends of the wipers?

Find the value of each variable.

5. 6.

7. A kite is attached to a 100 foot string as shown in the diagram. How far above the ground is the kite

when the string forms a 30° angle with the ground?

Check Point

Find a height Example 6

Find the height of an equilateral triangle Example 4

Find lengths in a 30°- 60°- 90° triangle Example 5

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Goals: Use the tangent ratio for indirect measurement.

A ____________________________________________ is a ratio of the lengths

of two sides in a right triangle. You will use trigonometric ratios to find the

measure of a side or an acute angle in a right triangle.

The ratio of the lengths of the legs in a right triangle is constant for a given angle

measure. This ratio is called the ______________________ of the angle.

Find tan 𝑆 and tan 𝑅. Write each answer as a fraction and as a decimal rounded to four decimal places.

Find tan 𝐽 and tan 𝐾. Round to four decimal places.

1. 2.

Section 7.5 – Apply the Tangent Ratio

Find tangent ratios Example 1

Check Point

Tangent Ratio

tan𝐴 =length of leg opposite ∠𝐴

length of leg adjacent to ∠𝐴=𝐵𝐶

𝐴𝐶

Let ∆𝐴𝐵𝐶 be a right triangle with acute ∠𝐴.

The tangent of ∠𝐴 (written as tan A) is defined as follows.

Remember these abbreviations: tangent → tan, opposite → opp, and adjacent → adj

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Find the value of x.

Find the height h of the lighthouse to the nearest foot.

Use a special right triangle to find the tangent of a 60° angle.

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

3. 4.

5. Find the height of the flagpole to the nearest foot.

Find a leg length

Example 2

Estimate height using tangent

Example 3

Use a special right triangle to find a tangent

Example 4

Check Point

14

Goals: Use the sine and cosine ratios.

The __________________ and _______________________ ratios are trigonometric ratios for acute angles that

involve the lengths of a leg and the hypotenuse of a right triangle.

Find sin 𝑆 and sin 𝑅. Write each answer as a fraction and as a decmal rounded to four places.

Find 𝐬𝐢𝐧 𝑿 and 𝐬𝐢𝐧 𝒀. Write each answer as a fraction and as a decmal. Round to four decimal places, if

necessary.

1. 2.

Find sine ratios

Example 1

Sine and Cosine Ratio

sin𝐴 =length of leg opposite ∠𝐴

length of hypotenuse=𝐵𝐶

𝐴𝐵

cos𝐴 =length of leg adjacent to ∠𝐴

length of hypotenuse=𝐴𝐶

𝐴𝐵

Let ∆𝐴𝐵𝐶 be a right triangle with acute ∠𝐴.

The sine of ∠𝐴 and cosine of ∠𝐴 (written as sin A and cos𝐴)

are defined as follows.

Remember these abbreviations: sine → sin, cosine → cos, hypotenuse → hyp

Section 7.6 – Apply the Sine and Cosine Ratios

Check Point

Check Point

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Find cos 𝑈 and cos 𝑊. Write each answer as a fraction and as a decimal (rounded to four decimal places).

You walk from one corner of a basketball court to the opposite corner. Write and solve

a proporton using a trignometric ratio to approximate the distance of the walk.

Find 𝐜𝐨𝐬 𝑹 and 𝐜𝐨𝐬 𝑺. Write each answer as a fraction and as a decmal. Round to four decimal places, if

necessary.

3. 4.

5. A rope, staked 20 feet from the base of a building, goes to the roof and forms an

angle of 58° with the ground. To the nearest tenth of a foot, how long is the rope?

Find cosine ratios

Example 2

Use a trigonometric ratio to find a hypotenuse

Example 3

Check Point

S

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Use a sine or cosine ratio to find the value of a and b.

Angles

If you look up at an object, the angle your line of sight makes with a horizontal line is alled the _____________

__________________________________. If you look down at an object, the angle your line of sight makes

with a horizontal line is call the _______________________________________________.

You are at the top of a roller coaster 100 feet above the ground. The angle of

depression is 44°. About how far do you ride down the hill?

A railroad crossing arm that is 20 feet long is stuck with an angle of elevation of 35°.

Find the lengths of x and y.

Use trig ratios to find the values of the variables

Example 4

Find a hypotenuse using an angle of depression

Example 5

Find a hypotenuse using an angle of elevation

Example 6

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Use a special triangle to find the sine and cosine of a 60° angle.

6. Use a sine or cosine ratio to find the value of e and f.

7. A pilot is looking at an airport from her plane. The angle of depression is

29°. If the plane is at an altitude of 10,000 feet, approximately how far is

it from the airport?

8. A dog is looking at a squirrel at the top of a tree. The distance between the two

animals is 55 feet and the angle of elevation is 64°. How high is the squirrel and

how far is the dog from the base of the tree?

9. Use a special triangle to find the sine and cosine of a 30° angle.

Use a special triangle to find a sine and cosine

Example 7

Check Point

18

Goals: Use inverse tangent, sine, and cosine ratios.

To solve a right triangle is to __________________________________________________________________

_________________________________________________________________________________________.

Use a calculator to approximate the measure of ∠𝐴 to the nearest tenth of a degree.

Let ∠𝐴 and ∠𝐵 be acute angles in a right triangle. Use a calculator to approximate the measures of ∠𝐴 and ∠𝐵

to the nearest tenth of a degree.

a) sin 𝐴 = 0.87 b) cos 𝐵 = 0.15

a) b) c)

Section 7.7 – Solve Right Triangles

Check Point

Use an inverse tangent to find an angle measure

Example 1

Use an inverse sine and inverse cosine Example 2

Inverse Trigonometric Ratios

Let ∠𝐴 be an acute angle.

Inverse Tangent: If tan𝐴 = 𝑥, then tan−1 𝑥 = 𝑚∠𝐴.

Inverse Sine: If sin𝐴 = 𝑦, then sin−1 𝑦 = 𝑚∠𝐴.

Inverse Cosine: If cos𝐴 = 𝑧, then cos−1 𝑧 = 𝑚∠𝐴.

Find the measure of ∠𝐴 to the nearest tenth of a degree Example 3

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Suppose your school is building a raked stage. The

stage will be 30 feet long from front to back, with a

total rise of 2 feet. A rake (angle of elevation) of 5° or

less is generally preferred for the safety and comfort of

the actors. Is the raked stage you are building within

the range suggested?

1. Use a calculator to approximate the measure of ∠𝑄 to the nearest tenth of a degree.

Let ∠𝑪 be an acute angle in a right triangle. Use a calculator to approximate the measures of ∠𝑪 to the

nearest tenth of a degree.

2. sin 𝐶 = 0.24 3. cos 𝐶 = 0.37

Find the measure of ∠𝑨 to the nearest tenth of a degree.

4. 5. 6.

7. You lean a ladder against a wall. The base of the ladder is 4 feet from the wall.

What angles 𝜃 does the ladder make with the ground?

Check Point

Solve a real-world problem Example 4

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Solve the right triangle formed by the water slide shown in the figure. Round

decimal answers to the nearest tenth.

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

8. 9.

10. You are standing 350 feet away from a skyscraper that is 750 feet tall. What is the angle of elevation

from you to the top of the building?

Solve a right triangle Example 5

Check Point