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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)]

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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)].

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