if a be the area of right triangle and b one of the sides containing the right angle.prove that the length of altitude on hypo is 2ab / underroot bsquare +4aquare
Answers
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)].