McGraw-Hill Education GRE 2019

1

1

45°

45°

2√

The legs of every isosceles right triangle will have a specific ratio that you should

memorize:

leg opposite 45 : leg opposite 45 : side opposite 90

↓ ↓ ↓

1 1 √ 2

x x x√2

It is important to note that the preceding combination only specifies a ratio, and

not actual values. For example, if you are told that the leg length of an isosceles

triangle is 5, then the hypotenuse is 5√ 2. Or if the leg length is 7, then the

hypotenuse is 7√ 2. The best way to think about the relationships of the leg lengths is

that the hypotenuse will be √ 2 × the leg.

One commonly tested fact about 45-45-90 triangles is that the diagonal of a

square will form two 45-45-90 triangles. This is helpful because you can use the

diagonal of the square to solve for the side lengths of the square and vice versa.

x x^2 x

x

x

√

What is the area of a square with a diagonal of length 20?

x^20

x

SOLUTION: To solve for the area, you need the length of a side.

The length of the side will be the leg of an isosceles right

triangle with a hypotenuse of 20. Let x = leg length:

x√2 = 20

x =

20

√2

area = x^2 =

20

√2^2 =

400

2 = 200

CHAPTER 13 ■ GEOMETRY 383

04-GRE-Test-2018_313-462.indd 383 12/05/17 12:04 pm