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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
.
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