Chapter 7Right Triangles and Trigonometry

7.1 Apply the Pythagorean Theorem

7.2 Use the Converse of the Pythagorean Theorem

7.3 Use Similar Right Triangles

7.4 Special Right Triangles

7.5 Apply the Tangent Ratio

7.6 Apply the Sine and Cosine Ratios

7.7 Solve Right Triangles

Name _________________________________ Block _______

SOL G.7

The student, given information in the form of a figure or statement, will prove two

triangles are similar, using algebraic or coordinate methods as well as deductive proofs.

PythagoreanTheorem

a

b

c

• Pythagorean Theorem is used to find the

third side of any ___________________________.

• Sides a and b are called the _____________.

• Side c is called the ____________________.

• For any right triangle,

__________________________.

Find the length of the unknown side of the right triangle. Determine whether the unknown side is

a leg or a hypotenuse.

1. 2.

3. 4.

5. A 16 foot ladder rests against the side of a

house, and the base of the ladder is 4 feet

away. Approximately how high above ground

is the top of the ladder?

6. Find the area of an isosceles triangle with

side lengths 10 meters, 13 meters, and 13

meters.

6

8

x

x3

5

x4

6

12

x9

7.1 and 7.2 – Pythagorean Theorem

Tell whether the given triangle is a right triangle.

7. 8.

9

15

3 34

26

22

14

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

9. 4, 4 3, and 8 10. 10, 11 and 14 11. 5, 6 and 61

Classifying TrianglesGiven a triangle with sides a, b and c …

• If __________________________, then the triangle is a ________________________ triangle.

• If __________________________, then the triangle is a ________________________ triangle.

• If __________________________, then the triangle is a ________________________ triangle.

12. 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?

13. Show that segments with lengths 3, 4 and 6 can form a triangle and classify the triangle as

acute, right or obtuse.

Keep c2 on the left!

You try!In #1-6, use ∆ABC to determine if the equation is true or false.

1. b2 + a2 = c2 4. c2 = a2 - b2

2. c2 – b2 = a2 5. c2 = b2 + a2

3. b2 – c2 = a2 6. a2 = c2 – b2

Find the unknown side lengths. Radical answers should be written in simplest form.

7. 8. 9.

Find the area of the figure. Write answers in simplest radical form when necessary.

10. 11. 12.

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

the triangle as right, acute or obtuse.

13. 5, 12 and 13 14. 8 ,4 and 6 15. 20, 21, and 28

A

B C

b

a

c

19

7

x

x

12

13 5

6

x

10 ft

7 ft

4 c

m

11 in

14 in

20 in

7.3 Similar Right Triangles

If the altitude is drawn to the hypotenuse of a right

triangle, then the two triangles formed are similar the

original and each other.

_________ ~ __________ , _________ ~ __________

and _________ ~ __________

Altitude Theorem Leg Theorem

Geometric Mean

altitu

de

Side 1

Altitude

Altitude

Side 2 =

Big ∆ leg

and little ∆ hypotenuse Big ∆ hyp

=Big ∆ leg

Little ∆ leg

Little ∆ hyp

Solve for the variable. Leave all answers in simplest radical form.

3. 4.R

S

Q

9

P

y

3

10

2

k

Video tutorials

for Geometric

Mean

Find the geometric mean of the two numbers.

1. 4 and 25 2. 6 and 20

*hint – look for the “T” shape* *hint – look for the “L” shape*

Complete and solve the proportions.

5. 6. 7.

= =x

12 8

12

8x

15 =x

x

x

15

20x

x

x9

9

11

You try!Find the value(s) of the variable(s).

1. 2.

3. 4.

Tell whether the triangle is a right triangle.

5. 6. 7.

Warm-Up: Discovering Special Right Triangles

1. What are the measures of ∠B and ∠C? Explain how you arrived at your answer.

2. Classify the triangle above by its angles and sides.

3. Write the Pythagorean theorem. Your equations will all be in terms of x and h because

the legs of the triangle are x units long and the hypotenuse is h units long.

4. Solve for h. Your answer should be in simplest radical form. (Think Algebra!!)

5. Redraw the triangle above, substituting your answer from step 3 for h in the diagram.

6. Summarize your findings in a complete sentence.

7.4 Special Right Triangles

˚ – ˚ – ˚

Special Right ∆45˚

Leg =

Hypotenuse =

45˚

The legs of a

45˚ – 45˚ – 90˚

triangle are

always

congruent.

a

a

a 2

Video tutorial

for special right

triangles

Review: Rationalizing the Denominator

Remove the radical from the denominator by multiplying the numerator and the denominator

by a fraction of 1.

1. 2 2. 15 3. 1

3 5 3 6

Directions: Find the length of the hypotenuse. Write all radical answers in simplest form. (No decimals)

1. 2.

45˚

83 2

3.

Directions: Find the length of the legs in the triangle. Write all radical answers in simplest form. (No decimals)

4. 5.

45˚

xx

2 2

6.

x

˚ – ˚ – ˚

Special Right ∆

30˚ Long leg =

Hypotenuse =

60˚

3

1

2

Short leg =

Short leg = Short Leg

LongLeg

Directions: Find the values of the variables. Write all radical answers in simplest form. (No decimals)

7. 8 9. The logo on a recycling bin

resembles an equilateral triangle with side lengths of 6 cm. What is the approximate height of the logo?

60˚

9

yx

30˚

x

3

x

Find the value of each variable. Write your answer in simplest radical form.

7. 8. 9.

You try!Find the value of x. Write your answer in simplest radical form.

1. 2. 3.

4. 5. 6.

45˚

8

45˚

9 2

x 4 2

45˚10

45˚

30˚

60˚

y

x3 3

yx

830˚

60˚

18

xy

30˚

60˚

x

xx

45˚

4

x

7.5 and 7.6 Trigonometry Ratios

Each acute angle of a right triangle has the following trigonometric ratios.Trigonometric RATIOS SINE

COSINE

TANGENT

The ratio of the leg

__________________ the angles to

the __________________.

The ratio of the leg

________________ to the angle to

the __________________.

The ratio of the leg

__________________ the angle to

the leg __________________ to the

angle.a

cb

A

C B

Sin A =

Sin B =

Cos A =

Cos B =

Tan A =

Tan B =

How to remember the

ratios!

SOH CAH TOA

Sin = Cos = Tan =

Directions: Find the sine, cosine, and tangent ratios for each acute angle in the triangle. Write

each answer as a decimal. Round to the nearest thousandths place when necessary.

1. 2.

178

15A

B

C

1665

63T

S

R

sin A: sin B:

cos A: cos B:

tan A: tan B:

sin A: sin B:

cos A: cos B:

tan A: tan B:

Video tutorial

for trig ratios

3. 4. 5.

Find the value of each variable. Round to the nearest hundredths place.

x

32˚ x y

18

51˚

yx

10

48˚

6. Jake leaned a 12 foot ladder against his house. If the angle formed by the ladder and the

ground is 68˚, how far from the back of the house did he place the ladder?

7. A ramp is used to load suitcases on an airplane. If the cargo door is 7 feet from the ground

and the angle formed by the end of the ramp and the ground is 25˚, how long is the ramp?

8. Casey sights the top of an 84 foot tall lighthouse at an angle of elevation of 58˚. If Casey is 6

feet tall, how far is he standing from the base of the lighthouse?

Angle of Elevation

Angle of Depression

Looking up at an object, then angle your

line of sight makes with a horizontal line

Looking down at an object, then angle your

line of sight makes with a horizontal line

Angle of elevation

Angle of depression

Hey

down

there!

Hey up

there!

9. A lifeguard is sitting on a platform, looking down at a swimmer in the water. If the lifeguard’s

line of sight is 8 feet above the ground and the angle of depression to the swimmer is 18˚,

how far away is the swimmer from the lifeguard?

You try!

1. sin30˚ 2. cos18˚ 3. tan72˚

4. tan42˚ 5. sin83˚ 6. cos65˚

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

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

10. A pilot in a helicopter spots a landing pad below. If the angle of elevation is 73˚ and the horizontal distance to the pad is 1200 feet, what is the altitude of the helicopter?

11. The angle of elevation form a kicker’s foot on the football field to the top of the goal post bars is 17. If he is standing 131 feet from the base of the goal post, how tall is the goal post?

7. 8. 9. y

x10

36˚

64˚x

y

8

8

yx

48˚

7.7 Solving Right Triangles

To solve a right triangle – to find the measures of all the sides and angles – you must know either:

a. two side lengths

b. one side length and one acute angle

Inverse Trigonometric

Ratios

Inverse Sine

Inverse Cosine

Inverse Tangent

If sinA = x, then sin-1x = m∠A

If cosA = y, then cos-1y = m∠A

If tanA = z, then tan-1z = m∠A

A C

B

** inverse ratios find the measure of the unknown ANGLE **

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

a. b.

2. 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. sinA = 0.87 b. cosB = 0.15

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

a. b.

15

20AB

C

14

11CB

A

70

BC

A

42˚

G H

J

14

16

Laws for Solving Triangles

Law of Sines Law of CosinesThis works for any non-right triangle when you are

given ___________ sides and the

______________________ angle OR two angles and the

____________________________ side.

This works for and non-right triangle when you are

given two sides and the ______________________

angle OR when you are given _______________ sides.

Video tutorialfor Law of Sinesand Law of Consines

1. Find BC. 2. Find AB. 3. Find m∠G.

Find the indicated measures.

4. For ∆ABC find the length of c given a = 14, b = 20, and m∠C = 120˚.

You try!Find the indicated angle measures.

1. Find m∠A. 2. Find m∠G. 3. Find m∠M.

4. Find m∠B. 5. Find m∠T. 6. Find m∠D.

A

B

C

3712

G H

I

8

6

M

N

O

10

26

A

B

C

24

32 R S

T

9

1317

D

E F

20

29

34

Find the indicated measures.

7. Find m∠T. 8. Find AB. 9. Find m∠B.

R

S

T

7

9

15

A B

C

15 12

42˚

18

13

38˚BA

C