2014 AMC 12B Problems/Problem 21
Problem 21
In the figure, is a square of side length
. The rectangles
and
are congruent. What is
?
Solution1
Draw the attitude from to
and call the foot
. Then
. Consider
. It is the hypotenuse of both right triangles
and
, and we know
and
, so we must have
.
Notice that all four triangles in this picture are similar and thus we have . This means
is the midpoint of
. So
, along with all other similar triangles in the picture, is a 30-60-90 triangle, and we have
and subsequently
. This means
, which gives
, so the answer is
.
Solution2
Let . Let
. Because
and
,
are all similar. Using proportions and the pythagorean theorem, we find
Because we know that
, we can set up a systems of equations
Solving for
in the second equation, we get
Plugging this into the first equation, we get
Plugging into the previous equation with
, we get
2014 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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