Question

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.

Show that the minimum length of the hypotenuse is (a^2/3 +b^2/3)^3/2

Answer 1

Let ∆ABC be right-angled at B. Let AB = x and BC = y. Let P be a point on the hypotenuse of the triangle such that P is at a distance of a and b from the sides AB and BC respectively.

Therefore, by second derivative test, the length of the hypotenuse is the maximum when

Hence, the maximum length of the hypotenuses is