Chapter 7 Chapter 7 Lesson 3Lesson 3

Objective:Objective: To use the properties of 30°-60°-90°

triangle.

Theorem 7-9:Theorem 7-9: 30°-60°-90° Triangle Theorem30°-60°-90° Triangle Theorem

In a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is √3 times the length of the shorter leg.

hypotenuse = 2 • shorter leg

longer leg = √3 • shorter leg

30°

60°lo

ng

leg

short leg

hypote

nus

e

30°

60°

x√3

x2x

Example 1:Example 1:Finding the Lengths of the LegsFinding the Lengths of the Legs

Find the value of each variable.

60°

30°

x

y

8Shorter Leg

hypotenuse = 2 • shorter leg

8 = 2x

x = 4

Longer Leglonger leg = √3 • shorter leg

y = x√3

y = 4√3

Example 2:Example 2:Finding the Lengths of the LegsFinding the Lengths of the Legs

Find the lengths of a 30°-60°-90° triangle with hypotenuse of length 12.

60°

30°

y

x12

Shorter Leghypotenuse = 2 • shorter leg

12 = 2x

x = 6

Longer Leglonger leg = √3 • shorter leg

y = x√3

y = 6√3

Example 3:Example 3:Finding the Lengths of the LegsFinding the Lengths of the Legs

60°

30°

x

y

4√3

Shorter Leghypotenuse = 2 • shorter leg

4√3 = 2x

x = 2√3

Longer Leglonger leg = √3 • shorter leg

y = x√3

y = 2√3•√3

Y=6

Find the lengths of a 30°-60°-90° triangle with hypotenuse of length 4√3.

Example 4:Example 4:Using the Length of a LegUsing the Length of a Leg

Find the value of each variable.

30°

60°

5

xy

Shorter Leglong leg = √3 • short leg

35 x

33

35

x

335

x

HypotenuseHyp. = 2 • shorter leg

xy 2

3

352 y

3310

y

Example 5:Example 5:Using the Length of a LegUsing the Length of a Leg

30°

60°

x

√6y

Longer Leglonger leg = √3 • shorter

leg63 x

18x

29 x

Hypotenusehyp. = 2 • shorter leg

xy 2

62 y

62y

The shorter leg of a 30°-60°-90° has length √6. What are the lengths of the other sides? Leave your answers in simplest radical form.

23x

Example 6:Example 6:Using the Length of a LegUsing the Length of a Leg

30°

60°

18

xy

Shorter Leg long leg = √3 • short leg

x 318

318

x

33

318

x

Hypotenusehyp. = 2 • shorter leg

362 y

312y

The longer leg of a 30°-60°-90° has length 18. Find the length of the shorter leg and the hypotenuse.

3318

x

36x

HomeworkHomework

Page 369 – 371

#12-22; 24-27; 30-33; 35;38