### All GMAT Math Resources

## Example Questions

### Example Question #11 : Calculating Whether Quadrilaterals Are Similar

In Quadrilateral , , , , is a right angle.

There exists Quadrilateral such that Quadrilateral Quadrilateral , and .

Which of the following is true about the areas of the two quadrilaterals?

**Possible Answers:**

Quadrilateral has area twice that of Quadrilateral .

Quadrilateral has area three times that of Quadrilateral .

Quadrilateral and Quadrilateral have the same area.

Quadrilateral has area three times that of Quadrilateral .

Quadrilateral has area twice that of Quadrilateral .

**Correct answer:**

Quadrilateral has area twice that of Quadrilateral .

We will assume that and have common measure 1 for the sake of simplcity; this reasoning is independent of the actual measure of .

The Quadrilateral with its diagonals is shown below. We call the point of intersection :

The diagonals of a quadrilateral with two pairs of adjacent congruent sides - a *kite* - are perpendicular; also, bisects the and angles of the kite. Consequently, is a 30-60-90 triangle and is a 45-45-90 triangle. By the 30-60-90 Theorem, since and are the short leg and hypotenuse of ,

.

By the 45-45-90 Theorem, since and are a leg and the hypotenuse of ,

The similarity ratio of Quadrilateral to Quadrilateral can be found by finding the ratio of the length of side to corresponding side :

The ratio of the areas is the square of the similarity ratio:

The correct choice is that Quadrilateral has area twice that of Quadrilateral .