Question

A triangle has sides A, B, and C. Sides A and B are of lengths `9` and `8`, respectively, and the angle between A and B is `pi/4`. What is the length of side C?

Answer 1

C = `sqrt(82)`

Explanation:

Note that `pi/4" radians "-> 45^o`

Let `/_cdb = pi/2 " that is: " 90^o`

As`/_ dcb = pi/4 " then " /_cbd=pi/4`

Thus it follows that `bd =dc=9`

Thus `ad = 9-8=1`

By Pythagoras: `(bd)^2+(da)^2 = C^2`

`=> C = sqrt( (bd)^2+(da)^2)`

`=> C = sqrt(9^2+1^2) =sqrt(82)`

Whilst 82 is not prime it is a product of prime numbers. As far as I am aware the only way of showing the precise value of C is to express it in this form.

Thus C = `sqrt(82)`