1

Name ____________________________________________________________ Period ______

CHAPTER 9 Right Triangles and Trigonometry

Section 9.1 Similar right Triangles

Objectives: Solve problems involving similar right triangles.

Use a geometric mean to solve problems.

Ex. 1 Use the diagrams at the right to review definitions of a right triangle.

1. Name the legs of ABC .

2. Name the hyotenuse of ABC .

3. What is the measure of A and C ?

4. Name the legs of DEF .

5. Name the hypotenuse of DEF .

6. Could DEF have an obtuse angle? Explain.

Theorem 9.1 If the altitude is drawn to the hypotenuse of a right triangle,

then the two triangles formed are similar to the original

triangle and to each other.

Ex. 2 Use the diagram.

a. Sketch the three similar triangles. Label the vertices.

b. Write similarity statements for the three triangles.

Remember that the geometric mean of two positive numbers a and b is the positive number x such

that a

xx

bWhen you write statements of proportionality, some sides appear in more than one triangle.

Shorter leg of CBD longer leg of CBD Shorter leg of ACD longer leg of ACD Shorter leg of ABC longer leg of ABC Shorter leg of CBD longer leg of CBD Shorter leg of ABC longer leg of ABC Shorter leg of ACD longer leg of ACD

The results are listed in the theorems on the next page.

BD

CDCD

AD

BD

CDCD

AD

BD

CDCD

AD

X

2

Theorem 9.2

In a right triangle, the altitude from the right angle to the hypotenuse

divides the hypotenuse into two segments.

The length of the altitude is the geometric mean of the lengths of the two segments.

Theorem 9.3 In a right triangle, the altitude from the right angle to the hypotenuse

Divides the hypotenuse into two segments.

The length of each leg of the right triangle is the geometric mean of the lengths

of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

Ex. 3 Complete and solve the proportion. 3. 9

xx

1. x

12

8 2.

15

xx

Ex. 4 Write similarity statements for the three similar triangles in the diagram. Then find the

given length.

4. Find QS. 5. Find TU.

Ex. 5 Find the value of each variable.

6. 7.

BD

CDCD

AD

AB

CBCB

DB

AB

ACAC

AD

3

Section 9.2 The Pythagorean Theorem

Objectives: Proving the Pythagorean Theorem

Using the Pythagorean Theorem

In this lesson, you will study one of the most famous theorems in mathematics – the Pythagorean

theorem. The relationship it describes has been known for thousands of years. No one know at

what point in history this relationship was first discovered. The ancient Babylonians and Chinese

were known to use this relationship.

THEOREM 9.4 Pythagorean Theorem:_______________________________

_________________________________________________________________

_________________________________________________________________

There are many different proofs. Here is a visual one.

A Pythagorean triple is ___________________________________________________________

_______________________________________________________________________________

Find the length of the hypotenuse of the right triangle. Tell whether the sides form a Pythagorean

triple.

Ex.1 Try this:

Find the length of the leg of the right triangle.

Ex. 2 Try this:

4

Ex 3 Two boats start at the same point on a lake. One heads north for 100 yards and the other heads west

for 50 yards. How far apart are the boats? Make a sketch first. Round to the nearest tenth.

Ex. 4 Find the area of the triangle to the nearest tenth of a meter.

Try this:

Section 9.3 The Converse of the Pythagorean Theorem

THEOREM 9.5 Converse of the Pythagorean:

_______________________________________________________________________________

_______________________________________________________________________________

_______________________________________________________________________________

Ex. 5 The triangle appears to be a right triangle. Is it a right triangle?

Try this:

Theorem 9.4 If △ABC is a right triangle, then c² = a² + b².

Theorem 9.5 If c² = a²+ b² , then △ABC is a right triangle. (Converse theorem)

Theorem 9.6 If c² < a² + b² , then ▵ ABC is an acute triangle.

Theorem 9.7 If c² > a² + b² , then ▵ ABC is an obtuse triangle.

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

classify the triangle as right, acute, or obtuse.

a. 1, 3, 5 b. 3, 5, 7 c. 10, 49, 50 d. 17, 144, 145

5

Section 9.4 Special Right Triangles

Objectives: Side lengths of Special Right Triangles

Special Right Triangles in real life.

Theorem 9.8 4 – 4 – 9 Triangle Theorem: ____________________________

_____________________________________________________________________

Theorem 9.9 3 – 6 – 9 Triangle Theorem: __________________________

____________________________________________________________________

____________________________________________________________________

Right triangl – – – –

special right triangles. If you are given one side, you can use the formulas to find the other two

sides. They can be proven by using the Pythagorean Theorem.

Find the value of each variable.

Ex. 1 Try this:

Ex. 2

Try this:

Ex. 3 (Ex 3 in the book) Find the values of s and t.

6

Ex 4 Use the diagram to tell whether the equation is true or false.

t 3h 7 h

2

Ex 5 a. The side lengths of an equilateral triangle is 5 cm. Find the length of an altitude of the

triangle. (Hint: Draw a sketch first.)

b. The perimeter of a square is 32 inches. Find the length of a diagonal.

Ex 6 A tipping platform is a ramp used to unload trucks. How high is the end of an 80 foot ramp

when

Ex. 7 Given the following information, what are the measures of the 2 acute angles?

Ex. 8 Find the area. Round to the nearest tenth.

a. b.

7

Section 9.5 Trigonometric Ratios

Objectives: Finding Trigonometric Ratio

Using trigonometric Ratios in Real Life

A trigonometric ratio is a _________________________________________________________

_______________________________________________________________________________ The word trigonometry is derived from the ancient Greek language and means measurement of triangles.

The three basic trigonometric ratios are sine, cosine. and tangent, which are abbreviated as sin, cos, and

tan. respectively. The saying sohcahtoa helps remember the ratios.

trigonometric ratio

Let ABC be a right triangle. The sine, the cosine, and the tangent of the

Acute angle A are defined as fallows.

sinA side opposite A

hypotenusea

c

cosA side adjacent A

hypotenuseb

c

tanA side opposite A

side adjacent Aa

b

Use the diagrams at the right to find the trigonometric ratio.

1. sin A 2. cos A

3. tan B 4. sin J

5. cos K 6. tan K

Find the sine, the cosine, and the tangent of the acute angles of the triangle. Express each value as

a decimal rounded to four places.

7. 8.

8

Use a calculator to approximate the given value to four decimal places. Set your calculator to

degrees.

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

15.

16.

17.

18. A train is traveling up a slight grade w traveling 1 mile

what is the vertical change in feet?

19

horizontally about 58.2 meters. Estimate the height h of the slide.

9

Section 9.6 Solving Right Triangles

Objectives: Solving a right triangle

Using right triangles in real life

☞ sohcahtoa ☜

Match the trigonometric expression with the correct ratio. Some ratios may be used more than

once, and some may not be used at all.

1. sin A = 2. cos A =

3. tan A = 4. sin B =

5. cos B = 6. tan B =

Once you know the sine, the cosine, or the tangent of an acute angle, you can use a calculator to

find the measure of the angle.

In general, for an acute angle A:

if sin A = x, then sin ¹x = mA.

if cos A = x, then cos ¹x = mA

if tan A = x, than tan ¹x = mA

A is an acute angle. Use a calculator to approximate the measure of A. Round to one decimal place.

7. sin A = 0.24 8. tan A = 1.73

9. cos A = 0.64 10. sin A = 0.38

Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To solve

a right triangle means to determine the measures of all six parts. You can solve a right triangle if

you know either of the following:

Two side lengths

One side length and one acute angle measure

Solve the right triangle. Round decimals to the nearest tenth.

11. 12.

10

13. A sonar operator on a ship detects a submarine at a distance of 400 meters and an angle of

14. The Uniform Federal Accessibility Standards specify that the ramp angle used for a

wheelchair e

vertical rise is E ’ z

Section 9.7 Vectors

Objective: Finding the Magnitude of a Vector

Match the vector with the correct component form of the vector.

1. 2. 3.

A. 2,4 B. 4,2 C. 2,4

The magnitude of a vector ABu ruu

is the distance from the initial point A to the terminal point B and

is written ABu ruu

.

Write the vector in component form. Find the magnitude of the vector. Round your answer tot he

nearest tenth.

4. 5. 6.