Question

A triangle has two corners with angles of `pi / 4` and `pi / 4`. If one side of the triangle has a length of `16`, what is the largest possible area of the triangle?

Answer 1

128

Explanation:

The angles of a triangle have to sum up to `pi` and since two of the angles are both `pi/4`, the remaining angle must be `pi/2`. Knowing special triangles, this is a right triangle where the two legs are the same length and the hypotenuse is `sqrt2` times the legs.

One of the sides of the triangle has a length of `16`. We can set that to be either the leg length or the hypotenuse length. Intuitively setting it as the leg length will give a larger triangle because the sides will be longer: `16, 16, and 16sqrt2~~22.6` instead of `16/sqrt2~~11.3, 16/sqrt2~~11.3, and 16`.

Since our largest triangle has the largest area, we know that the first set is what we're looking for.

Since this is a right triangle with legs 16 and 16, the area is `16^2/2=128`