Question

If A be the area of a right angled triangle and x is one of the sides containing right angle. Prove that the length of altitude on the hypotenuse is 2Ax/√x^4+4A^2

Answer 1

Answer:

Step-by-step explanation:

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)].