Question

Two corners of a triangle have angles of `pi / 6` and `pi / 12`. If one side of the triangle has a length of `8`, what is the longest possible perimeter of the triangle?

Answer 1

45.314cm

Explanation:

The three angles for triangle are `pi/6, pi/12 and 3/4pi`

To get the longest perimeter, the shortest length shall reflex to the smallest angle.
Let say that the other lengths are b reflex to angle `pi/6` and c reflex to angle `3/4pi` while a=8 reflex to angle `pi/12`

therefore
`a/sinA =b/sinB =c/sinC`

`b/sin(pi/6) =8/sin(pi/12)`

`b=8/sin(pi/12)*sin(pi/6)`
`b=8/0.2588*0.5`
`b=15.456`

`c/sin((3pi)/4) =8/sin(pi/12)`

`c=8/sin(pi/12)*sin((3pi)/4)`
`c=8/0.2588*0.7071`
`c=21.858`

The longest possible perimeter = a+b+c
`=8+15.456+21.858`
`= 45.314cm`