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 midpoint of AC and BD.

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

\[BO = OD = \frac{1}{2}BD = 7 cm\]

\[{AB}^2 + {AD}^2 = 2 {AO}^2 + 2 {BO}^2 \left( \text{by Apollonius theorem} \right)\]
\[ \Rightarrow 130 = 2 {AO}^2 + 2 \left( 7 \right)^2 \]
\[ \Rightarrow 130 = 2 {AO}^2 + 2 \times 49\]
\[ \Rightarrow 130 = 2 {AO}^2 + 98\]
\[ \Rightarrow 2 {AO}^2 = 130 - 98\]
\[ \Rightarrow 2 {AO}^2 = 32\]
\[ \Rightarrow {AO}^2 = 16\]
\[ \Rightarrow AO = 4 cm\]

Since point O is the midpoint of side AC.

∴ AC = 2AO = 8 cm

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