If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between them is .

Given: a right-angled triangle, such that sum of the lengths of its hypotenuse and side is given

To show: the area of the triangle is maximum when the angle between them is

Explanation:

Let ΔABC be the right-angled triangle,

Let hypotenuse, AC = y,

side, BC = x, AB = h

Now sum of the side and hypotenuse is given,

⇒ x+y = k, where k is any constant value

⇒ y = k-x………..(i)

Now let A be the area of the triangle, then we know

Now applying the Pythagoras theorem, we get

y² = x²+h²

Now substituting equation (i) in above equation, we get

Now substituting the value from equation (iii) into equation (ii), we get

Now differentiating above equation with respect to x, we get

Now taking out the constant term we get

Now applying the product rule, we get

Applying the power rule of differentiation on second part in above equation, we get

Now critical point is found by equating the first derivative to 0, i.e.,

⇒k²-2kx = kx

⇒k² = 2kx+kx

⇒ k² = 3kx

⇒ k = 3x

Again, differentiating equation (iv) with respect to x, we get

Taking out the constant term and taking the LCM, we get

Applying the product rule of differentiation, we get

Now applying the power rule of differentiation, we get

Substituting , in above equation, we get

Hence the maximum value of A is at

We know,

Now from figure,

Substituting the value of y = k-x from equation (i), we get

Substituting the value of , we get

This is possible only when

Hence the area of the triangle is maximum only when the angle between them is