Chapter 9 Summary

Similar Right Triangles

• If the altitude is drawn to the hypotenuse of a right triangle, then the 3 triangles are all similar.

A B

C

D

altitudepart

partaltitude 2

1

Find QS.

Solve for x

A B

C

D

leghypotenuse

adjacentleg

Solve for x

Find XZ

Pythagorean Theorem

• In a right triangle,

222 cba

Acute, Right, Obtuse Triangles

• Acute

• Right

• Obtuse

222 bac

222 bac

222 bac

Pythagorean Triples

• Any 3 whole numbers that satisfy the pythagorean theorem.– Example: 3, 4, 5– Nonexample: anything with a decimal or square

root!

45°-45°-90° Triangle Theorem

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

x

xx√2

45°

45°

Hypotenuse = √2 leg∙

30°-60°-90° Triangle Theorem

• In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

x√3

60°

30°

Hypotenuse = 2 shorter leg∙Longer leg = √3 shorter leg∙

2xx

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle

• Find the value of x• By the Triangle Sum Theorem, the measure of the third angle

is 45°. The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.

3 3

x

45°

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle

Hypotenuse = √2 ∙leg

x = √2 3∙ x = 3√2

3 3

x

45°

45°-45°-90° Triangle Theorem

Substitute values

Simplify

Ex. 2: Finding a leg in a 45°-45°-90° Triangle

• Find the value of x.• Because the triangle is

an isosceles right triangle, its base angles are congruent. The triangle is a 45°-45°-90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg.

5

x x

Ex. 2: Finding a leg in a 45°-45°-90° Triangle5

x x

Statement:Hypotenuse = √2 leg∙

5 = √2 x∙

Reasons:45°-45°-90° Triangle Theorem

5

√2

√2x

√2=

5

√2x=

5

√2x=

√2

√2

5√2

2x=

Substitute values

Divide each side by √2

Simplify

Multiply numerator and denominator by √2

Simplify

Ex. 3: Finding side lengths in a 30°-60°-90° Triangle

• Find the values of s and t.• Because the triangle is a 30°-60°-90° triangle,

the longer leg is √3 times the length s of the shorter leg.

5

st

30°

60°

Ex. 3: Side lengths in a 30°-60°-90° Triangle

Statement:Longer leg = √3 shorter leg∙

5 = √3 s∙

Reasons:30°-60°-90° Triangle Theorem

5

√3

√3s

√3=

5

√3s=

5

√3s=

√3

√3

5√3

3s=

Substitute values

Divide each side by √3

Simplify

Multiply numerator and denominator by √3

Simplify

5

st

30°

60°

The length t of the hypotenuse is twice the length s of the shorter leg.

Statement:Hypotenuse = 2 shorter leg∙

Reasons:30°-60°-90° Triangle Theorem

t 2 ∙

5√3

3= Substitute values

Simplify

5

st

30°

60°

t 10√3

3=

Parts of the TriangleSohCahToa

oppositehypotenuse

Sin OppHyp

adjacent

Cos AdjHyp

Tan OppAdj

hypotenuseopposite

adjacent

Find the values of the three trigonometric functions of .

4

3

opphyp

45

adjhyp

35

oppadj

43

sin cos tan

5

Finding a missing side

Finding a missing angle

• We can find an unknown angle in a right triangle, as long as we know the lengths of two of its sides.–Use Trig Inverse–sin-1

–cos-1

–tan-1

What is sin-1 ?But what is the meaning of sin-1 … ?

Well, the Sine function "sin" takes an angle and gives us the ratio “opposite/hypotenuse”

But in this case we know the ratio “opposite/hypotenuse” but want to know the angle.

So we want to go backwards. That is why we use sin-1, which means “inverse sine”.

Find the missing angle