Question

Two corners of a triangle have angles of `pi / 6` and `pi / 2`. If one side of the triangle has a length of `3`, what is the longest possible perimeter of the triangle?

Answer 1

`9+3sqrt(3)`

Explanation:

The longest perimeter will occur if the given side length is the shortest side length i.e. if 3 is the length opposite the smallest angle, `pi/6`

By definition of `sin`
`color(white)("XXX")3/h=sin(pi/6)`

`color(white)("XXX")rarr h=3/sin(pi/6)=3/(1/2)=6`

Using the Pythagorean Theorem
`color(white)("XXX")x=sqrt(6^2-3^2)=sqrt(27)=3sqrt(3)`

Perimeter `=3+h+x=3+6+3sqrt(3)=9+3sqrt(3)`