# Parallelogram with sides as 7 and 9, and Integer diagonals. Find sum of all possible products of lengths of diagonals?

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A given parallelogram has sides measuring 7 and 9, and both its diagonals have integer lengths. Find the sum of all possible products of the lengths of the diagonals.

posted Aug 28, 2018

14&8

a=7, b=9, d1=?, d2=?
As per the cosine theorem the formula of parallelogram diagonal in terms of sides and cosine α:
d1 = sqrt(a^2 + b^2 + 2ab*cosα)=sqrt(130+126*cosα) --> (1)
d2 = sqrt(a^2 + b^2 - 2ab*cosα) = sqrt(130-126*cosα) --> (2)
Let's consider α between 0 & 90 deg, excluding 0 (at 0 deg the parallelogram becomes a line), where cosα gains a value between 0 & 1. Similar result will be gained when α=90-180 deg.
Since both d1 & d2 are integers, 130+126*cosα will be equal to a number between 130 & 256 and 130-126*cosα will be equal to a number between 4 & 130. For (1) we can try 225, 196, 169, 144 and for (2)- 35, 64, 91, 116.
The only possible pair of integers are d1=14, d2=8

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