Answer
$5\sqrt{2} \text{ } cm$
Work Step by Step
The diagonal and 2 sides of the square form a right triangle. Let $c$ be the hypotenuse and $a,b$ be the equal legs. Using $a^2+b^2=c^2$ or the Pythagorean Theorem, where $c=10$ and $a=b,$ then
\begin{array}{l}\require{cancel}
a^2+b^2=c^2
\\\\
a^2+a^2=10^2
\\\\
2a^2=100
\\\\
a^2=\dfrac{100}{2}
\\\\
a^2=50
\\\\
a=\sqrt{50}
\\\\
a=\sqrt{25\cdot2}
\\\\
a=\sqrt{(5)^2\cdot2}
\\\\
a=5\sqrt{2}
.\end{array}
Hence, the side of the square (which has the same value as the legs) is equal to $
5\sqrt{2} \text{ } cm
.$
