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?

1 Answer
Mar 26, 2017

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#