What is the area of a rhombus of side 13 such that the sum of its two diagonals is 34?

Question

The lengths of two adjacent sides of a parallelogram are 8 and 15, respectively. Find the sum of squares of the lengths of the two diagonals.

Answer 1

We can write two equations:
(1) d1 + d2 = 34
(2) (d1/2)^2 + (d2/2)^2 = 13^2 or d1^2 + d2^2 = 676
Solving the system of two equations.
We can easily obtain: 2d2^2 - 68 d2 + 480 = 0
Two possible solutions for d2 : 24 and 10
If d2 = 24 , d1 = 10 If d2 = 10, d1 = 24
In order to calculate the area of the rhomdus, both solutions are valid.
Area of the rhombus = (diagonal 1 * diagonal 2) / 2 = 120
The answer is 120