2022 AMC 10B Problems/Problem 16
Problem
The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
Solution
Label the points on the diagram.
By doing some angle chasing, using the fact that and
are right angles, we find that
,
, and
are congruent. Similarly,
,
, and
are congruent as well.
Therefore, .
Since the triangles are created by a rectangle and a square, and
.
is a right angle, so
.
is also
, and using the similar triangles,
and
.
by subtracting
from
. The area of a 3-4-5 right triangle is
, so both
and
have an area of 6. The ratio of
to
=
, so by similarity, the area of of
is
.
and
, so the area of rectangle
is
. Subtracting the area of
and
, gets the shaded area,
~Connor132435