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प्रश्न
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.
उत्तर
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.
\[{AB}^2 + {AD}^2 = 2 {AO}^2 + 2 {BO}^2 \left( \text{by Apollonius theorem} \right)\]
\[ \Rightarrow 130 = 2 {AO}^2 + 2 \left( 7 \right)^2 \]
\[ \Rightarrow 130 = 2 {AO}^2 + 2 \times 49\]
\[ \Rightarrow 130 = 2 {AO}^2 + 98\]
\[ \Rightarrow 2 {AO}^2 = 130 - 98\]
\[ \Rightarrow 2 {AO}^2 = 32\]
\[ \Rightarrow {AO}^2 = 16\]
\[ \Rightarrow AO = 4 cm\]
Since point O is the midpoint of side AC.
Hence, the length of the other diagonal is 8 cm.
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