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?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{given}\mathrm{quadrilateral}\mathrm{be}\mathrm{ABCD}\mathrm{in}\mathrm{which}\mathrm{diagonals}\mathrm{AC}\mathrm{is}\mathrm{equal}\mathrm{to}6\mathrm{cm}\mathrm{and}\mathrm{BD}\mathrm{is}\mathrm{equal}\mathrm{to}8\mathrm{cm}. \\ \mathrm{Also},\mathrm{it}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{the}\mathrm{diagonals}\mathrm{bisect}\mathrm{each}\mathrm{other}\mathrm{at}\mathrm{right}\mathrm{angle},\mathrm{at}\mathrm{point}\mathrm{O}. \\ \therefore \mathrm{AO}=\mathrm{OC}=\frac{1}{2}\mathrm{AC}=3\mathrm{cm} \\ \mathrm{Also},\mathrm{OB}=\mathrm{OD}=\frac{1}{2}\mathrm{BD}=4\mathrm{cm} \\ \mathrm{In}\mathrm{right}△\mathrm{AOB}: \\ {\mathrm{AB}}^2={\mathrm{OA}}^2+{\mathrm{OB}}^2 \\ \Rightarrow {\mathrm{AB}}^2=(9+16){\mathrm{cm}}^2 \\ \Rightarrow {\mathrm{AB}}^2=25{\mathrm{cm}}^2 \\ \Rightarrow \mathrm{AB}=5\mathrm{cm} \\ \mathrm{Thus},\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{each}\mathrm{side}\mathrm{of}\mathrm{the}\mathrm{quadrilateral}\mathrm{is}5\mathrm{cm}. \end{gathered}$$