Question

In the figure above, $ABCD$ is a rectangle and $FC=ED$. What fraction of the rectangle is shaded?

• A. $0.5$
• B. $0.4$
• C. $0.6$
• D. $0.8$
• E. $0.9$

Answer 1

The correct option is A $0.5$

Area of shaded portion = $A\left(\square ABCD\right)-A\left(\Delta BFG\right)-A\left(\Delta AEG\right)-A\left(\Delta DEH\right)-A\left(\Delta CFH\right)$ where points $G$ and $H$ are respectively where the shaded portion intersects the sides $AB$ and $CD$.

$=AD\times AB-\frac{1}{2}\times BF\times BG-\frac{1}{2}\times AG\times AE-\frac{1}{2}\times DH\times DE-\frac{1}{2}\times CH\times CF$

Since it is given that $FC=ED$, it also implies that $BC-FC=AD-ED$ i.e. $BF=AE$

Area of the shaded region thus becomes $AD\times AB-\frac{1}{2}\times BF\times (BG+AG)-\frac{1}{2}\times CF\times (DH+CH)$

$=AD\times AB-\frac{1}{2}\times BF\times AB-\frac{1}{2}\times CF\times CD$

$=AD\times AB-\frac{1}{2}\times AB\times (BF+CF)$

$=AD\times AB-AB\times \frac{BC}{2}$

$=AD\times AB\times \frac{1}{2}$

Thus the shaded region equals half of the area of the rectangle.