2013 AMC 8 Problems/Problem 24
Problem
Squares ,
, and
are equal in area. Points
and
are the midpoints of sides
and
, respectively. What is the ratio of the area of the shaded pentagon
to the sum of the areas of the three squares?
Solution 1 (shortcut)
It can be proven that (where
is the point where
intersects
) which also means quadrilaterals
(due to the squares being equal in area which means the squares are congruent, and since the triangles earlier mentioned are congruent). The area of the shaded region is equal to the area of one square since the quadrilaterals and triangles are congruent. The total area of the shape is the area of three squares. Putting these two pieces of information together,
~ julia333