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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?
Quadrilateral and Quadrilateral have the same area.
Quadrilateral has area twice that of Quadrilateral .
Quadrilateral has area three times that of Quadrilateral .
Quadrilateral has area twice that of Quadrilateral .
Quadrilateral has area three times that of Quadrilateral .
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 .