Question

Hey guys.. Solve this one Plzzz......
If A be the area of a right triangle and b be one of the sides containing the right angle, prove that the length of the altitude on the hypotenuse is 2Ab divided by underroot b4+4Asq.

Answer 1

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)].