Question

A parallelogram has sides with lengths of `15` and `12`. If the parallelogram's area is `24`, what is the length of its longest diagonal?

Answer 1

Given: A parallelogram with sides `a= 15`, `b = 12`, and `"Area" = 24`

We know the formula for the area of a para

`"Area" = bh" [1]"`

We know that

`sin(theta) = h/a" [2]"`

Solve equation [2] for h:

`h = asin(theta)" [2.1]"`

Substitute equation [2.1] into equation [1]:

`"Area" = (ab)sin(theta)`

`sin(theta) = "Area"/(ab)`

Substitute in the known values:

`sin(theta) = 24/(15(12))`

`sin(theta) = 2/15`

`theta = sin^-1(2/15)`

This angle is opposite the shortest diagonal.

The angle opposite the longest diagonal is;

`alpha = pi-sin^-1(2/15)`

The diagonal, d, opposite `alpha` can be found using the law of cosines:

`d = sqrt(a^2+ b^2 - 2(a)(b)cos(alpha)`

`d ~~ 26.94`