Problem:
In rectangle ABCD,AB=1,BC=2, and points E,F, and G are midpoints of BC,CD, and AD, respectively. Point H is the midpoint of GE. What is the area of the shaded region?
Answer Choices:
A. 121​
B. 183​​
C. 122​​
D. 123​​
E. 61​
Solution:
Let J be the intersection point of BF and HC. Then △JHF is similar to △JCB with ratio 1:2. The length of the altitude of △JHF to HF plus the length of the altitude of △JCB to CB is FC=21​. Thus △JHF has altitude 61​ and base 1, and its area is 121​. The shaded area is twice the area of △JHF, or (E)61​​.
OR
Place the figure on the coordinate plane with H at the origin. Then the equation of line DH is y=2x, and the equation of line AF is y=−4x−1. Solving the equations simultaneously shows that the leftmost point of the shaded region has x-coordinate −61​. The kite therefore has diagonals 31​ and 1, so its area is 21​⋅31​⋅1=(E)61​​.
