Warmup:Solve for y

y

2y

12

8.2 Special Right Triangles

LEQ: What are special right triangles and how do we use them?

The “root” of it all…Calculate the value of each hypotenuse in a

triangle as a reduced radical…

Do you notice a trend? Is there an easier way to calculate (shortcuts)?

(hint: label sides with numbers)

Side lengths of Special Right Triangles• Right triangles whose angle

measures are 45°-45°-90° or 30°-60°-90° are called special right triangles.

• The theorems that describe these relationships of side lengths of each of these special right triangles follow….

Theorem 8-5: 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 = leg ∙ √2

Finding the hypotenuse in a 45°-45°-90° Triangle

Ex 1: Find the value of x at the right.

Ex 2: Find the length of the hypotenuse of a 45-45-90 triangle with legs of length 5√3.

2√16

x

45°

Finding a leg in a 45°-45°-90° Triangle

Ex. 3 Find the value of x and y.

5

x y

45°

Theorem 8-6: 30°-60°-90° Triangle Theorem• In a 30°-60°-90°

triangle:– the hypotenuse

is twice as long as the shorter leg

– the longer leg is √3 times as long as the shorter leg.

x√3

60°

30°

Hypotenuse = 2 ∙ (shorter leg)

Longer leg = (shorter leg) ∙ √3

2xx

Alternatives? OF COURSE!!

If you don’t care for shortcuts, you may memorize…

1

1√21

2

√3

30

60

45

45

Finding side lengths in a 30°-60°-90° Triangle

Ex. 1 Find the values of s and t and the right.

Ex. 2 Find the value of each variable.

5

st

30°

60°

30°

60°8

x

y

Real World Connection

The yield sign is an equilateral triangle. It has a height of 18√3.

Find x and y.

18√3

y inch.

x inch.