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Prove that -

+2 votes
asked Aug 21, 2016 in Mathematics by Rahul Roy (7,955 points)
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)]

1 Answer

+2 votes
answered Aug 21, 2016 by vikash (21,287 points)
selected Sep 4, 2016 by sarthaks
Best answer

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)].

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