McGraw-Hill Education GRE 2019

30-60-90 Triangles and the Equilateral Triangle

The other type of special right triangle you need to master is the 30-60-90 triangle.

To understand the properties of a 30-60-90 triangle, look at what happens when

you draw the height of an equilateral triangle:

60 °60°

6

6 6

60°

→

60 °60°

33

6 30°30° 6

33√

Fact 1: The height of an equilateral triangle will cut the base in half.

Fact 2: The resulting smaller triangles will have degree measurements of

30-60-90.

Fact 3: The sides of the 30-60-90 triangle will be in the following ratio,

which you must memorize:

60°

30°

2(3) = 6

3

3 √ 3

30:60:90

1 :√3:2

x:x√3:2x

Finally, as was the case with 45-45-90 triangles, with

30-60-90 triangles, the side relationships only specify ratios,

not values.

60°

45°

30°

A

B D

25√ C

What is the perimeter of triangle BCD in the figure above?

SOLUTION: To determine the perimeter, you must determine the side lengths of

BCD. Note that side BC is the hypotenuse of the 45-45-90 triangle ABC. Thus

BC = 5√ 2 × √ 2 = 5 × 2 = 10. Since BC is the shorter leg of 30-60-90 triangle

BCD, the longer leg, CD, will equal 10√ 3 , and the hypotenuse, BD, will equal

10 × 2 = 20. The perimeter of BCD is thus 10 + 20 + 10√ 3 = 30 + 10√ 3.

60°

45°

10

20

30°

A

B D

25√ C

310√

384 PART 4 ■ MATH REVIEW

04-GRE-Test-2018_313-462.indd 384 12/05/17 12:04 pm