# Two corners of a triangle have angles of # (2 pi )/ 3 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 4 #, what is the longest possible perimeter of the triangle?

##### 1 Answer

#### Explanation:

The problem gives you two out of the three angles in an arbitrary triangle. Since the sum of the angles in a triangle must add up to 180 degrees, or

Let's draw the triangle:

The problem states that one of the sides of the triangle has a length of 4, but it does not specify which side. However, in any given triangle, it is true that the *smallest* side will be opposite from the smallest angle.

If we want to maximize the perimeter, we should make the side with length 4 the side opposite from the smallest angle. Since the other two sides will be larger than 4, it guarantees that we will maximize the perimeter. Therefore, out triangle becomes:

Finally, we can use the **law of sines** to find the lengths of the other two sides:

Plugging in, we get:

Solving for x and y we get:

Therefore, the maximum perimeter is:

**Note:** Since the problem does not specify the units of length on the triangle, just use "units".