Question

How do you find perimeter of a triangle with 1 side given the angles are 105, 45, and 30 and the only side given is 8, opposite of the angle 30?

Answer 1

Perimeter: `34.76852` (to the limit of my calculator)

Explanation:

Let `a` be the length of the side opposite the `45^@` angle
and `b` the length of the side opposite the `105^@` angle.

The Law of Sines tells us
`color(white)("XXX")8/sin(30^@)=a/sin(45^@)=b/sin(105^@)`

`sin(30^@)=0.5`
`sin(45^@)=0.707107` (approx.)
`sin(105^@)=0.965926` (approx.)

`8/sin(30^@)=16`

`a=16xxsin(45^@)=11.31371`

`b=16xxsin(105^@)=15.45481`

`"Perimeter" = 8 + a + b`

`color(white)("XXXXX")=34.76852`