Question

In rectangle $ABDF$ above, $C$ and $E$ are midpoints of sides $\overline{BD}$ and $\overline{DF}$, respectively. What fraction of the area of the rectangle is shaded?

• A. $\frac{5}{2}$
• B. $\frac{5}{3}$
• C. $\frac{5}{4}$
• D. $\frac{5}{8}$

Answer 1

The correct option is D $\frac{5}{8}$

Let the side of rectangle $ABDF$ is $AB$ and $BD$ is $x$ and $y$ respectively.

Then area of rectangle $ABDF=xy$

The point $C$ is mid point of $BD$ and $E$ the mid point of $DF$

Then $BC=CD=$ $\frac{y}{2}$ and $DE=$ $\frac{X}{2}$

Then area of triangle $ABC=$ $\frac{1}{2}\times x\times \frac{y}{2}=\frac{xy}{4}$

And area of triangle $CDF=$ $\frac{1}{2}\times \frac{x}{2}\times \frac{y}{2}=\frac{xy}{8}$

Then area of shaded rectangle $=$ $xy-\frac{xy}{4}-\frac{xy}{8}=\frac{8xy-2xy-xy}{5}=\frac{5}{8}xy$

Then fraction of the shaded rectangle and rectangle $=$ $\frac{\frac{5}{8}xy}{xy}=\frac{5}{8}=0.625$