CHAPTER 7

7.1 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

a2 + b2 = c2

a, leg

b, leg

c, hypotenuse

PYTHAGOREAN TRIPLE a set of 3 positive integers a, b, and c

that satisfy the equation a2 + b2 = c2.

3,4,55,12,13 8,15,17 7,24,25and multiples of these numbers like…. 6,8,10 10,24,26 16,30,34

14,48,50

EXAMPLES: FIND X.

x

12

5

SOLVE FOR X.

x

2 14

2 5

A ramp for a truck is 6 feet long. The bed of the truck is 3 feet above the ground. How long is the base of the ramp?

FIND THE AREA OF THE TRIANGLE

24in

20in. 20in.

PROVING THE PYTHAGOREAN THEOREM

b

b

b

b

a

a

a

a c

c

c

c

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 triangle ABC is a right triangle.

OTHER THEOREMS FROM 7.2 If c2 < a2 + b2 , then the triangle is

acute.

If c2 > a2 + b2 , then the triangle is obtuse.

PROVE: IF C2 < A2 + B2 , THEN THE TRIANGLE IS ACUTE

cb

a

A

C B

Given Diagram:

xb

a

P

R Q

PROVE: IF C2 > A2 + B2 , THEN THE TRIANGLE IS OBTUSE. Given Diagram:

cb

a

A

C B

xb

a

P

R Q

7.3 USING SIMILAR RIGHT TRIANGLES Theorem - 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.

If CD is an altitude of ABC, then CBD ~ ABC ~ ACD

A

C

BD

EXAMPLE: Identify the similar triangles

D

EF

G

EXAMPLE: Side view of a tool shed What is the maximum height of the

shed to the nearest tenth?

17ft

9ft

15ft8ft

GEOMETRIC MEAN (ALTITUDE) THEOREM

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 length of the two segments

A

C

BD

GEOMETRIC MEAN (LEG) THEOREM 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.

A

C

BD

EXAMPLE: Find K

k

102

7.4 SPECIAL RIGHT TRIANLGES 45o-45o-90o Triangle Theorem-

In a 45o-45o-90o triangle, the hypotenuse is √2 times as long as each leg.

EXAMPLE: FIND X.

EXAMPLE: FIND X.

30O-60O-90O TRIANGLE THEOREM In a 30o-60o-90o triangle, the hypotenuse

is twice as long as the shorter leg and the longer leg is √3 times as long as the shorter leg.

EXAMPLE:

EXAMPLE: A logo in the shape of an equilateral

triangle Find the height of the logo. Each side is 2.5 inches long.

7.5 APPLYING THE TANGENT RATIO Trigonometric Ratio- a ratio of the

lengths of two sides of a right triangle. SOH – CAH- TOA

b

ac

B

CA

Tangent- ratio of the length of the opposite leg to the adjacent leg of a right triangle (Round to 4 decimal places.)- “TOA”

60

7545

D

E F

EXAMPLE Find x. Round to the nearest tenth.

17X

9

EXAMPLE Find the height of the flagpole to the

nearest foot.

65

24ft

EXAMPLES

7.6 APPLY THE SINE AND COSINE RATIOS

Sine “SOH”

Cosine “CAH”

b

ac

B

CA

EXAMPLES

EXAMPLE Solve the right triangle formed by the

water slide.

50ft42

Z Y

X

INVERSE TRIGONOMETRIC RATIOS Inverse tan tan-1

If tan A = x then, tan-1 x = m<A Inverse sin sin-1

If sin A = y, then sin-1 y= m<A Inverse cos cos-1

If cos A = z, then cos-1 z = m<A

b

ac

B

CA

EXAMPLE Angle C is an acute angle in a right

triangle. Approximate the m<C is to the nearest tenth degree when:

Sin C = 0.2400

Cos C = 0.3700

EXAMPLE Approximate the measure of angle Q to

the nearest tenth of a degree.

12

8Q S

R

EXAMPLE A road rises 10 feet over a 200 foot

horizontal distance. Find the angle of elevation.

EXAMPLES

7.7 SOLVING RIGHT TRIANGLES To solve a right triangle means to find

the measures of all of the sides and angles.

You need: 2 side lengths or one side and one angle

ANGLES OF ELEVATION AND DEPRESSION

Angle of Elevation- the angle your line of sight makes with a horizontal line while looking up

Angle of Depression- the angle your line of sight makes with a horizontal line while looking down

angl e of el evati on

angl e of depressi on

EXAMPLE You are skiing down a mountain with an

altitude of 1200m. The angle of depression is 21o. How far do you ski down the mountain? Round to the nearest meter.

EXAMPLE You are looking up at an airplane with an

altitude of 10,000ft. Your angle of elevation is 29o. How far is the plane from where you are standing? Round to the nearest foot.

PERIMETER EXAMPLE

15

120 20

UNIT CIRCLE

r

P(x,y)