**Question:**

The diagonals of a quadrilateral are of lengths 6 cm and 8 cm. If the diagonals bisect each other at right angles, what is the length of each side of the quadrilateral?

**Solution:**

Let the given quadrilateral be $\mathrm{ABCD}$ in which diagonals $\mathrm{AC}$ is equal to $6 \mathrm{~cm}$ and $\mathrm{BD}$ is equal to $8 \mathrm{~cm}$.

Also, it is given that the diagonals bisect each other at right angle, at point $\mathrm{O}$.

$\therefore \mathrm{AO}=\mathrm{OC}=\frac{1}{2} \mathrm{AC}=3 \mathrm{~cm}$

Also, $\mathrm{OB}=\mathrm{OD}=\frac{1}{2} \mathrm{BD}=4 \mathrm{~cm}$

In right $\triangle \mathrm{AOB}$ :

$\mathrm{AB}^{2}=\mathrm{OA}^{2}+\mathrm{OB}^{2}$

$\Rightarrow \mathrm{AB}^{2}=(9+16) \mathrm{cm}^{2}$

$\Rightarrow \mathrm{AB}^{2}=25 \mathrm{~cm}^{2}$

$\Rightarrow \mathrm{AB}=5 \mathrm{~cm}$

Thus, the length of each side of the quadrilateral is $5 \mathrm{~cm}$.