Question

Two corners of a triangle have angles of `(3 pi ) / 4` and `pi / 12`. If one side of the triangle has a length of `5`, what is the longest possible perimeter of the triangle?

Answer 1

Largest possible perimeter 28.3196

Explanation:

Sum of the angles of a triangle `=pi`

Two angles are `(3pi)/4, pi/12`
Hence `3^(rd)`angle is `pi - ((3pi)/4 + pi/12) = pi/6`

We know`a/sin a = b/sin b = c/sin c`

To get the longest perimeter, length 2 must be opposite to angle `pi/12`

`:. 5/ sin(pi/12) = b/ sin((3pi)/4 = c / sin (pi/6)`

`b = (5 sin((3pi)/4))/sin (pi/12) = 13.6603`

`c =( 5* sin(pi/6))/ sin (pi/12) = 9.6593`

Hence perimeter `= a + b + c = 5 + 13.6603 + 9.6593= 28.3196`