CHAPTER EIGHTAlec RodriguezJack WellsChris “the Bottman” Bott

8.1 Similarity in Right Triangles Theorem 8-1 Right Triangle Similarity If an 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.

C

A

B

D

Geometric Mean The mean between two numbers in a geometric

sequence.

2,4,8,16,32

a/x =x/b

Ex. 2/x = x/32

Answer: 8

Corollary 1 When the altitude is drawn to the

hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.

C

A

B

D AD/CD = CD/BD

Corollary 2 When the altitude is drawn to the

hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

C

A

B

D AB/AC= AC/AD

AB/BC = BC/BD

Challenge

C

A

B

D

Find AB, AC, CD, CB

9

16X

Y

Z

Triangle ABC is a right triangle.

The Pythagorean Theorem In a right triangle, the

square of the hypotenuse is equal to the sum of the squares of the legs.

A2 + B2 = C2

Pythagoras.

Challenge

6

8

X

1. Find X. 2. Find C.

45°

90°

90°

2√2

C

8-3 Converse of the Pythagorean Theorem

BA

CC2 = A2 + B2 Right Triangle

C2 < A2 + B2 Acute Triangle

C2 > A2 + B2 Obtuse Triangle

8-3 Converse of the Pythagorean Theorem

Example: Is the triangle acute, obtuse, or right? 132 + 152 ____ 292

169 + 225 ____ 841394 < 841The triangle is acute.

15

13

29

122 + 182____ 192

144 + 324 ____ 361468 > 361The triangle is obtuse.

1812

19

45

Special Right Triangles1. 45 – 45 – 90 General Rule

a

a a 2

45

More Special Right Triangles

2. 30 – 60 – 90General Rule

a

a 3

2a60

30

Even More Special Right Triangles Challenge

745

X

Find X.

30

4 3

YFind Y.

Sine Formula : sinѲ=Opposite Hypotenuse

Ѳ

Hypotenuse

Adjacent

Oppo

site

• Solve for x:

– Sin20=4/x– Multiply each

side by x– X(sin20)=4– Divide each

side by sin20– X=11.695

4 20⁰x

Cosine Formula : cosѲ=Adjacent Hypotenuse Solve for x:

Cos67=x/120 Multiply both sides by 120 120(cos67)=x Multiply 120 and cos67 46.88=x

Ѳ

Hypotenuse

Adjacent

Oppo

site x12

067⁰

Tangent Formula : tanѲ=Opposite Adjacent

Ѳ

Hypotenuse

Adjacent

Oppo

site

• Solve for x

• Tan42=x/5• Multiply each side by 5• 5(tan42)=x• Multiply 5 and tan42• 4.5=x

5

x

42⁰

SOH-CAH-TOA An easy way to remember all of these

formulas is by using SOH CAH TOA SOH - (sine) opposite over hypotenuse CAH - (cosine) adjacent over hypotenuse TOA - (tangent) opposite over adjacent

Applications of Right Triangle Trigonometry

How to solve:1. Tan2⁰ = 25/x2. x = 25/tan2⁰ 3. x = 716.3

Angle of elevation

Angle of depression

horizontal

horizontal

Line of sight

2⁰

2⁰

25

x

Exercises Solve for x and y:

x

y

3537⁰

When the sun’s angle of elevation is 57⁰, a building casts a shadow 21m long. How high is the building?

21m

57⁰

Last Exercise An observer is located 3km from a rocket

launch site sees a rocket at an angle of elevation of 38⁰. How high is the rocket at that moment?