Question

Points E and F are the midpoints of sides $AD$ and $BC$ of rectangle ABCD, point G is the intersection of $AF$ and $BE$, and point H is the intersection of $CE$ and $DF$ as shown in the above figure. The interior of ABCD except for the interior EGFH is shaded. Calculate the ratio of the area of EGFH to the area of the shaded region?

• A. $1:2$
• B. $1:3$
• C. $1:4$
• D. $1:6$
• E. Cannot be determined from the given information

Answer 1

The correct option is C $1:4$

$△AFE\equiv △BFG\equiv △EHD\equiv △GHC$ (ASA)

All four triangles are congruent.

Hence,

Area of these triangles are$=4\times \frac{lb}{2\times 2}=lb$

Similarly, $△BFA\equiv △FGE\equiv △GHE\equiv △CHD$

Hence,

Area of these triangles are $=4\times \left(\frac{l}{2}\right)\times b\times \frac{1}{2}$

$=lb$

Area of rectangle$=2lb$

Area of EFGH$=2\times \frac{l}{2}\times b\times \frac{1}{2}=\frac{lb}{2}$

Ratio$=\frac{lb}{2}\times \frac{1}{2lb}=1:4$