Question

If A be the area of a right triangle and ‘a’ one the sides containing the right angle, prove that the length of the altitude on the hypotenuse is 2Aa /√a4+4A2

Answer 1

Let us consider

Area of the right angle = A

One side of the right angle = b

Area of the triangle A = 1/2 × b × h

h = 2A/b

Another side of the right-angled triangle containing the right angle (h) = 2A/b

According to Pythagoras theorem, the hypotenuse of the right-angled triangle is,

(Hypotenuse)2=(Base)2+(Height)2

(Hypotenuse)2 = (b)2 + (2A / b)2

⇒ (Hypotenuse)2 = b2 + (4A2 / b2)

⇒ Hypotenuse = √[b2 + (4A2 / b2)

⇒ Hypotenuse = √[(b4 + 4A2) / b2

⇒ Hypotenuse = 1/b √[(b4 + 4A2)

According to Pythagoras theorem, area of the right angle is

A = 1/2 × 1/b √[(b4 + 4A2)] × altitude on hypotenuse (h)

⇒ 2A = 1/b √[(b4 + 4A2)] × h

⇒ 2Ab = √[(b4 + 4A2)] × h

⇒ h = 2Ab/√(b4 + 4A2)