Prove that -

Question

If A be the area of a right triangle and b be one of the sides containing the right angle, prove that the lenght of the altitude on the hypotenuse is 2Ab/[sq.rt of(b^4 + 4A^2)]

Answer 1

Sol:
Base of the right angled triangle is 'b' units.
Area of the right angled triangle is "A' sq units.
A = 1/2 × b × h
⇒ h = 2A / b

Another side of the right angled triangle containing the right angle = 2A / b

Hypotenuse of the right angled triangle according to Pythagoras theorem:

(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)]


Area of the right angle considering hypotenuse as the base.

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

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

⇒ 2Ab = √[(b⁴ + 4A²)] × altitude on hypotenuse

⇒ Altitude on hypotenuse = 2Ab / √[(b⁴ + 4A²)]

Therefore, length of the altitude on hypotenuse of the right angled triangle is 2Ab / √[(b⁴ + 4A²)].