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

Show that the maximum length of the hypotenuse is

Let ΔABC 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.

Let <C = θ .

Now, we have,

Ac =

Now, PC = b cosecθ

And AP = a secθ

⇒ AC = AP + PC

⇒ AC = a secθ + b cosecθ

Now, if

⇒ asecθ tanθ = bcosecθ cotθ

⇒

⇒ asin³θ =bcos³θ

⇒

⇒

………..(1)

So, it is clear that < 0 when

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

Now, when , we get,

Ac =

Therefore, the maximum length of the hypotenuses is .