Question

A parallelogram has sides 12cm and 18cm and a contained angle of 78 degrees. Find the shortest diagonal?

Answer 1

The length of the shorter diagonal is `~~19.45"cm"`.

Explanation:

Let the sides of your parallelogram be

`a = 12 "cm"` and `b = 18 "cm"`.

and your angle be

`gamma = 78^@`, the angle between `a` and `b` (or `b` and `a`)

As two adjacent angles in a parallelogram are suplementary, we know that the adjacent angle to `gamma` is

`beta = 180^@ - gamma = 180^@ - 78^@ = 102^@`

Now, we can use the law of cosines to compute both diagonals in the parallelogram.

Let `d` be the diagonal opposite to `gamma` and `e` be the diagonal opposite to `beta`.

The law of cosines states:

`d^2 = a^2 + b^2 - 2ab * cos(gamma)`

`e^2 = a^2 + b^2 - 2ab * cos(beta)`

Thus, we can compute the lengths of the diagonals as follows:

`d^2 = 12^2 + 18^2 - 2 * 12 * 18 * cos(78^@)`

`= 144 + 324 - 432 * cos(78^@)`

`~~ 468 - 432 * 0.208`

`~~ 378.18`

`=> d ~~ 19.45 "cm"`

And for the other diagonal,

`e^2 = 12^2 + 18^2 - 2 * 12 * 18 * cos(102^@)`

`= 468 - 432 * (-0.208)`

`~~ 557.82`

`=> e ~~ 23.62 "cm"`

Thus, the length of the shorter diagonal is `~~19.45"cm"`.