# Area of shaded region to the square

In this question, I was able to make out that root(2) * EF = 1/2 AD. Then area of the small square =

root(2) EF = 1/2 AD

=> EF = AD/2 root(2)

=> EF = AD * AD / 8

Therefore, ratio = AD*AD/8*AD*AD. The answer is B, however. What am I doing wrong?

• This can't be right, the picture suggests that the size of the inner square is about $1/3$ the size of the larger one. Where is this question from? – Orest Bucicovschi Jan 14 '15 at 13:00

## 1 Answer

First of all, the diagonal is not $\overline{AD}$ but instead $\overline{AC}$. Then just by moving the inner square to have a corner at $E\to A$, you can see that the small square must have half the side lengths as the big square, so its area is $\frac14$ of the big square, thus the shaded region has area $\frac34$ of the big square as claimed.

Your calculation is wrong because the texts wording contains an error. The proper equation would be $$\overline{EG} = \frac12 \overline{AC} \Rightarrow \overline{EF} = \frac1{\sqrt2} \overline{EG} = \frac1{2\sqrt2} \overline{AC} = \frac12 \overline{AB}$$