Problem 816

First method:

!

From the area of square $I J K L$, we can calculate the length of its side. Let the length of the side of square IJKL be $y$. From the condition of the problem, we have:

$y^{2}=200$, so $y=10 \sqrt{2} \text{ cm}$.

2 POINTS

Let the length of the side of square EFGH be $a$. Let $x=\frac{a}{2}$. Then we have:

$y=x \sqrt{2}$, so $10 \sqrt{2}=x \sqrt{2}$, which means $x=10 \text{ cm}$.

2 POINTS

The length of the side of square EFGH is $a=20 \text{ cm}$.

1 POINT

The ratio of the legs of the right triangle $A F E$ is $3: 4$. Let $|E A|=3 k$ and $|A F|=4 k$.

By the Pythagorean theorem, we have $(3 k)^{2}+(4 k)^{2}=20^{2}$,

so $25 k^{2}=400$, which means $k^{2}=16, k=4$.

The length of the side of square $A B C D$ is $|A F|+|F B|=|A F|+|E A|=16+12=28 \text{ cm}$.

##