GMAT Math : Calculating whether quadrilaterals are similar

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Example Questions

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Example Question #461 : Problem Solving Questions

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  and Quadrilateral  have the same area.

Quadrilateral  has area three times that of Quadrilateral .

Quadrilateral  has area three times that of Quadrilateral .

Quadrilateral  has area twice 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 .

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