Question

Two corners of a triangle have angles of `(2 pi )/ 3` and `( pi ) / 6`. If one side of the triangle has a length of `8`, what is the longest possible perimeter of the triangle?

Answer 1

The longest perimeter is `P ~~ 29.856`

Explanation:

Let `angle A = pi/6`
Let `angle B = (2pi)/3`

Then `angle C = pi - A - B`

`C = pi - pi/6 - (2pi)/3`

`C = pi - pi/6 - (2pi)/3`

`C = pi/6`

Because the triangle has two equal angles, it is isosceles. Associate the given length, 8, with the smallest angle. By coincidence, this is both side "a" and side "c". because this will give us the longest perimeter.

`a = c = 8`

Use the Law of Cosines to find the length of side "b":

`b = sqrt(a^2 + c^2 - 2(a)(c)cos(B))`

`b = 8sqrt(2(1 - cos(B)))`

`b = 8sqrt(2(1 - cos((2pi)/3)))`

`b = 8sqrt(3)`

The perimeter is:

`P = a + b + c`

`P = 8 + 8sqrt(3) + 8`

`P ~~ 29.856`