# GMAT Math : Calculating whether quadrilaterals are similar

## Example Questions

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### 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?

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 .

Quadrilateral has area twice that of Quadrilateral .

Explanation:

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 .

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