# 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?

Mar 26, 2017

128

#### Explanation:

The angles of a triangle have to sum up to $\pi$ and since two of the angles are both $\frac{\pi}{4}$, the remaining angle must be $\frac{\pi}{2}$. Knowing special triangles, this is a right triangle where the two legs are the same length and the hypotenuse is $\sqrt{2}$ 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: $16 , 16 , \mathmr{and} 16 \sqrt{2} \approx 22.6$ instead of $\frac{16}{\sqrt{2}} \approx 11.3 , \frac{16}{\sqrt{2}} \approx 11.3 , \mathmr{and} 16$.

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 ${16}^{2} / 2 = 128$