Prove that of all triangles having two given sides of length

Prove that of all triangles having two given sides of lengths a and b, the one whose sides form a right angle encompasses the maximum possible area.

Solution

Let x be the angle between sides a and b.

The area of the triangle with two sides a and b with angle x between them is A = 1/2 * b * a * sin(x) since if b is the base of the triangle then sin(x) = h / a and h = a * sin(x)

We graph y = sin(x) from 0 to 90 degrees. sin(0) = 0, and as x increases sin(x) increases until it hits its maximum of sin(90) = 1

If sin(x) is at a maximum when x = 90, then A = 1/2 * b * a * sin(x) will also be at a maximum when x = 90. If x = 90, then a and b form a right angle.

Therefore, a right angled triangle has the maximum area for two given side lengths