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Ch 8: Right Triangles and Trigonometry

8-1 The Pythagorean Theorem and Its Converse

8-2 Special Right Triangles

8-3 The Tangent Ratio

8-4 Sine and Cosine Ratios

8-5 Angles of Elevation and Depression

8-1: The Pythagorean Theorem and Its Converse

Focused Learning Target: I will be able to

• Use the Pythagorean Theorem

• Use the Converse of the Pythagorean

Theorem.

CA Standard(s):

Geo 15.0 Students use the Pythagorean theorem to

determine distance and find missing lengths of sides of

right triangles.

Vocabulary:

• Pythagorean Triple

Although the sides of a right triangle will always satisfy the Pythagorean Theorem, Pythagorean triples are a

special set of positive whole numbers that satisfy the equation 2 2 2

a b c+ = .

Examples of common Pythagorean Triples: Not Pythagorean Triples:

3, 4, 5 8, 15, 17 7, 10, 12.5

4

13,

4

3,

2

1

5, 12, 13 7, 24, 25 -3, -4, 5 1, 2, 3

Finding the length of a side using the Pythagorean Theorem and determining if it forms a Pythagorean triple:

I’ll do one:

Find the missing side, and then determine if it forms a Pythagorean triple with the other two sides.

We’ll do one together:

Find the missing side(s), and then determine if all three sides form a Pythagorean triple.

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You Try:

Find the missing side(s) and determine if it forms a Pythagorean triple with the other two sides.

Simplifying Radicals:

Using Simplest Radical Form:

When solving for a variable by taking the square root, the value is often

not a perfect square. In those cases it is often best to leave your

answer as a radical in simplest form. Here is one method:

1) Divide the number by 2, 3, 5, 6, etc (skip the perfect squares)

2) When the result is a perfect square you can simplify by taking the

square of the perfect square. Leave the other factor inside the radical.

If it does not factor in this way, then it is already in simplest radical

form.

I’ll do one: We’ll do one: You try one:

Simplify:

75

Simplify:

726

Simplify:

980

Examples: Using the Pythagorean Theorem and Simplifying Radicals

I’ll do one:

If a = 2, b = 4, find c. Leave your answer in simplest radical form.

The first 20 Perfect Squares chart 2

1 = 1 211 = 121

22 = 4 2

12 = 144 2

3 = 9 213 = 169

24 = 16 2

14 = 196 2

5 = 25 215 = 225

26 = 36 2

16 = 256 2

7 = 49 217 = 289

28 = 64 2

18 = 324 2

9 = 81 219 = 361

210 = 100 2

20 = 400

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We’ll do one together:

Find x. Leave your answer in simplest radical form.

You Try:

If a = 6, c = 12, find b. Leave your answer in simplest radical form.

Examples: Applying the Pythagorean Theorem in real-world situations

I’ll do one:

The Parks Department rents Paddle boats at docks near each entrance to the park. To the nearest meter, how

far is it to paddle from one dock to the other?

We’ll do one together:

A highway detour affects a company’s delivery route. The plan showing the old route and the detour is below.

How many extra miles will the trucks travel once the detour is established? Round your answer to the nearest

tenth of a mile.

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You Try:

A bike messenger has just been asked to make an additional stop. Instead of biking straight from the law office

to the court, she is going to stop at City Hall in between. Using the picture below, determine how many

additional miles she will need to travel. Leave your answer in simplest radical form.

Using the converse to determine if it is a right triangle:

I’ll do one:

Is the triangle below a right triangle? Explain.

We’ll do one together:

Is the triangle below a right triangle? Explain.

You Try:

Is the triangle below a right triangle? Explain.

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8-2 Special Right Triangles

Focused Learning Target: I will be able to

• Use the properties of

45 45 90° − ° − ° triangles.

• Use the properties of

30 60 90° − ° − ° triangles.

CA Standard(s):

Geo 15.0: Students use the Pythagorean theorem to determine

distance and find missing lengths of sides of right triangles.

Geo 20.0: Students know and are able to use angle and side

relationships in problems with special right triangles, such as 30,

60, 90 triangles and 45, 45, and 90 triangles.

There are two special triangles that are often encountered. In this section, we will explore the characteristics

that are unique to these triangles.

The 45 45 90° − ° − ° triangle is an isosceles right triangle. It has all of the properties of an isosceles triangle and

all of the properties of a right triangle. The theorem below is often used as a quicker way to find missing

lengths of these special right triangles compared to the Pythagorean Theorem.

Finding the hypotenuse

I’ll do one: We’ll do one together: You try:

Find the value of the variable:

Find the value of the variable:

Find the value of the variable:

Finding the length of a leg:

I’ll do one:

Find the length of a leg of a 45 45 90° − ° − ° Triangle with a hypotenuse of length 10.

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We’ll do one together:

Find the value of the variables:

You try:

Find the value of the variable:

Another type of special triangle is the 30 60 90° − ° − ° triangle for which we use the theorem below:

I’ll do one:

Find the lengths of the missing sides.

We’ll do one together:

Find the missing variables:

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You Try:

Find the missing variables:

Some Real-World Situations

I’ll do one:

The moose warning sign below is an equilateral triangle. It is one meter high. Find the lengths of each of the

sides.

We’ll do one:

After heavy winds damaged a farmhouse, workers placed a 6-m brace against its side at a 45° angle. Then, at

the same spot on the ground, they placed a second brace at °30 angle.

a) How long is the longer brace? Round your answer to the nearest tenth of a meter.

b) About how much higher on the house does the longer brace reach than the shorter brace?

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You try:

Jefferson Park sits on one square city block 300 ft on each side. Sidewalks connect the opposite corners.

a) Draw the diagram

b) About how long is each diagonal sidewalk?

8-3 The Tangent Ratio

Focused Learning Target: I will be able to:

• Use the tangent ratios to

determine side lengths in triangles

CA Standard(s):

Geo 18.0: students know the definitions of the basic

trigonometric functions defined by the angles of a right triangle.

Geo 19.0: Students use trigonometric functions to solve for an

unknown length of a side of a right triangle, given an angle and a

length of a side.

Vocabulary:

Tangent: the tangent of acute A∠ in a right triangle is the ratio of the length of the leg opposite of A∠ to the

length of the leg adjacent to A∠ .

Example 1: Writing tangent ratios:

I’ll do one: We’ll do one: You try:

Write the tangent ratios for

T∠ and U∠ .

Write the tangent ratios for

E∠ and F∠ .

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Example 2: Real-World-Connection

I’ll do one:

We’ll do one:

Find the distance from the boathouse on shore to the cabin on the island.

You try:

Find x to the nearest whole number.

Example 3: Using the Inverse of Tangent,

−

adj

opp1tan , to find the missing angle

I’ll do one: We’ll do one: You Try:

Find the measure of angle X to the

nearest degree.

Find the value of x to the nearest

degree.

Find the value of x to the nearest

degree.

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8-4 Sine and Cosine Ratios

Focused Learning Target:

• Use sine and cosine to determine side lengths in triangles

CA Standard(s):

Geo 18.0: Students know the definitions of the basic trigonometric functions defined by the angles of a right

triangle. They are able to use elementary relationships between them.

Geo 19.0: Students use trigonometric functions to solve for an unknown length of a side of a right triangle,

given an angle and a length of a side.

Vocabulary:

Example 1: Writing the Sine and Cosine Ratios

I’ll do one:

We’ll do one:

Write the cosine and sine ratios for X∠ and Y∠ .

You Try:

Write the ratios for sin P and cos Q.

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How can we remember the ratios of Sine, Cosine, and Tangent? SOH, CAH, TOA!!!!

Hyp

OppangleSin =)( �

H

OS = � SOH

Hyp

AdjangleCos =)( �

H

AC = � CAH

Adj

OppangleTan =)( �

A

OT = � TOA

Example 2: Real-World Connection

I’ll do one:

We’ll do one:

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

You Try:

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

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Example 3: Using the Inverse of Sine,

−

hyp

opp1sin and Cosine,

−

hyp

adj1cos

I’ll do one:

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

We’ll do one:

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

You try:

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

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8-5 Angles of Elevation and Depression

Focused Learning Target:

• Use angles of elevation and

depression to solve problems.

Vocabulary:

• Angle of elevation

• Angle of depression

CA Standard(s):

Geo 18.0: Students know the definitions of the basic

trigonometric functions defined by the angles of a right triangle.

Geo 19.0: Students use trigonometric functions to solve for an

unknown length of a side of a right triangle, given an angle and

a length of a side.

The angle of elevation is the angle formed by taking the

horizontal line (as if you’re looking straight ahead) and then

looking upwards.

The angle of depression is the angle formed by taking the

horizontal line and then looking downwards.

In the picture to the right, notice that the angle of

depression from the person in the hot air balloon is

congruent to the angle of elevation from the person on the

ground. Why are those angles congruent?

Example 1: Identifying angles of elevation and depression

I’ll do one:

If you were at the peak of the mountain at

1∠ , what type of angle would that be; an

angle of elevation or depression?

We’ll try some together:

What about at 2∠ ? 3∠ ?

You try:

What about at 4∠ ?

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Example 2: Finding lengths in real-world scenarios using trigonometry

I’ll do one:

You are hiking and you spot a rock climber on a nearby cliff at a 32o angle of elevation. The horizontal ground

distance to the base of the cliff is 1000 ft. Find the height of the rock climber. (hint: always try to draw the

situation if none is given)

We’ll do one together:

To approach runway 17 of the Ponca City Municipal Airport in Oklahoma, the pilot must begin a 3o descent

starting from an altitude of 2714 ft. The airport’s altitude is 1007 ft above sea level. How many feet from the

runway is the airplane at the start of this approach? Then convert your answer into miles.

You try:

A coast guard helicopter pilot sights a life raft at a 26o angle of depression. The helicopter’s altitude is 3 km.

What is the helicopter’s horizontal distance from the raft?