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 `5`, what is the longest possible perimeter of the triangle?

Answer 1

The longest possible perimeter is, `p = 18.66`

Explanation:

Let `angle A = pi/6`

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

Then `angle C = pi - angle A - angle B`

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

`angle C = pi/6`

To obtain the longest perimeter, we associate the given side with the smallest angle but we have two angles that are equal, therefore, we shall use the same length for both associated sides:

side `a = 5` and side `c = 5`

We may use the Law of Cosines to find the length of side b:

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

`b = sqrt(5^2 + 5^2 - 2(5)(5)cos((2pi)/3)`

`b = 5sqrt(2 - 2cos((2pi)/3)`

`b = 5sqrt(2 - 2cos((2pi)/3)`

`b ~~ 8.66`

The longest possible perimeter is, `p = 8.66 + 5 + 5 = 18.66`