Question

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

Answer 1

Let ABCD be the given

parallelogram and its diagonals AC and BD intersect at point M.

∴ AB² + AD² = 130cm, BD = 14cm

MD = 1/2 BD (i) [Diagonals of a parallelogram bisect each other]

= 1/2 × 14 = 7 cm

In ∆ABD, seg AM is the median. [From (i)]

∴ AB^{​​​​​​​2} + = 2AM^{​​​​​​​2} + 2MD^{​​​​​​​2} [Apollonius theorem]

∴ 130 = 2 AM^{​​​​​​​2} + 2(7)^{​​​​​​​2} 

∴ 65 = AM^{​​​​​​​2} +49 [Dividing both sides by 2]

∴ AM² = 65 – 49

∴ AM² = 16 [Taking square root of both sides]

∴ AM = √16

= 4cm

Now, AC =2 AM [Diagonals of a parallelogram bisect each other]

2 × 4 = 8 cm

∴ The length of the other diagonal of the parallelogram is 8 cm.