Question

Sum of the squares of adjacent sides of a parallelogram is 130 sq.cm and length of one of its diagonals is 14 cm. Find the length of the other diagonal.

Answer 1

It is given that,
AB² + AD² = 130 sq. cm
BD = 14 cm

Diagonals of a parallelogram bisect each other.
i.e. O is the mid point of AC and BD.

In ∆ABD, point O is the midpoint of side BD.

$\mathrm{BO}=\mathrm{OD}=\frac{1}{2}\mathrm{BD}=7\mathrm{cm}$

$$\begin{gathered} {\mathrm{AB}}^2+{\mathrm{AD}}^2=2{\mathrm{AO}}^2+2{\mathrm{BO}}^2\left(\mathrm{by}\mathrm{Apollonius}\mathrm{theorem}\right) \\ \Rightarrow 130=2{\mathrm{AO}}^2+2{\left(7\right)}^2 \\ \Rightarrow 130=2{\mathrm{AO}}^2+2\times 49 \\ \Rightarrow 130=2{\mathrm{AO}}^2+98 \\ \Rightarrow 2{\mathrm{AO}}^2=130-98 \\ \Rightarrow 2{\mathrm{AO}}^2=32 \\ \Rightarrow {\mathrm{AO}}^2=16 \\ \Rightarrow \mathrm{AO}=4\mathrm{cm} \end{gathered}$$

Sinec, point O is the midpoint of side AC.

$\therefore \mathrm{AC}=2\mathrm{AO}=8\mathrm{cm}$

Hence, the length of the other diagonal is 8 cm.