Question

A parallelogram has sides with lengths of `9` and `8`. If the parallelogram's area is `32`, what is the length of its longest diagonal?

Answer 1

Longest diagonal `color(blue)(d_1 = 16.55)` units

Explanation:

Length of longest diagonal is given by

`d_1 = sqrt (a^2 + b^2 + 2a b cos theta)`

Where a and b are the two pairs of parallel lines and `theta` the acute angle between non parallel sides

To find `theta` first.

`Area` `A_p = a b sin theta`

Given `a = 9, b = 8, A_p = 32`

`sin theta=32 / (9 * 8) = 4/9`

`theta = sin ^-1 (4/9) = 0.4606`

`Longest diagonal`d_1 = sqrt(9^2 + 8^2 + (2 * 9 * 8 * cos 0.4606))#

`color(blue)(d_1 = 16.55` units