Special Right Triangles

Essential Question

How do I find the side lengths of special right triangles?

Activator

Solve. Assume all variables are positive.

1. c2 = 62 + 62

2. c2 – 42 = 42

3. a2 + 82 = 256

6 2

4 2

8 3

Definition

Special Right Triangles – right triangles whose angle measures are 45°-45°-90° or 30°-60°-90°.

45

4560

30

45°-45°-90° Triangle Theorem

x 2

45

45

x

xhypotenuse = leg 2

In a 45 -45 -90 triangle, the hypotenuse is2 times as long as each leg.

30°-60°-90° Triangle Theorem

In a 30 -60 -90 triangle, the hypotenuse istwice as long as the shorter leg, and thelonger leg is 3 times as long as theshorter leg.

x2x

x 3

60

30

hypotenuse = 2 shorter leg

longer leg = shorter leg 3

Example 1

Find the value of x. 45

x

33

hypotenuse = leg 2

x = 3 2

You Try!

Find the value of x. 45

x

55

hypotenuse = leg 2

x = 5 2

Example 2 Find the value of x.

5

xx

hypotenuse = leg 2

5 = x 2

5

2=

x 2

2

5

2=x

2

2 5

2=x

5 2

2=x

You Try!

Find the value of x. 12

xx

hypotenuse = leg 2

12 = x 2

12

2=

x 2

2

12

2=x

2

2 12

2=x

12 2

2=x

x = 6 2

Example 3

Find the values of s and t.

st

5

60

30

longer leg = shorter leg 3

5 = s 3

5

3=s

s = 5 3

3

hypotenuse = 2 shorter leg

t=25 3

3

t=10 3

3

You Try!

Find the values of f and g.

fg

18

60

30

longer leg = shorter leg 3

18 = f 3

18

3=f

f = 18 3

3

f = 6 3

hypotenuse = 2 shorter leg

g=2 6 3

g=12 3

Summarizer

Can the side lengths of a 45°-45°-90° triangle form a Pythagorean Triple?

No. The hypotenuse is 2 times the length ofthe leg. Therefore no triple of three integerscan be formed.