Question

A parallelogram has sides with lengths of `18` and `5`. If the parallelogram's area is `42`, what is the length of its longest diagonal?

Answer 1

Longest diagonal `color(crimson)(d_1 = 22.47` units

Explanation:

Length of the longest diagonal is given by the formula

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

a is one pair of parallel lines, b the other pair and theta the angle between non-parallel sides.

Given : `a = 18, b = 5, A_p = 42`

`a * b * sin theta = Area = A_p`

`sin theta = 42 / (18 * 5) = 1/2`

`theta = sin ^-1 (1/2) = pi/6`

Longest diagonal `d_1 = sqrt(18^2 + 5^2 + (2 * 18 * 5 * cos (pi/3)` units

`color(crimson)(d_1 = 22.47` units

Shortest diagonal `d_2 = sqrt(a^2 + b^2 - (2 a b cos theta)`

`d_2 = sqrt(18^2 + 5^2 - ( 2*18*5*cos(pi/3)`

`d_2 = 13.9` units