Prove that the length of the altitude on the hypotenuse, Mathematics

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If A be the area of a right triangle and b one of the sides containing the right angle, prove that the length of the altitude on the hypotenuse is 2 Ab /√ b4+4A2.

1300_triangel 4.png

Ans:    Let QR = b

A = Ar(ΔPQR)

A = 1/2 x b x PQ

PQ = 2A/b ...........................(1)

Δ PNQ ∼ ΔPQR (AA)

⇒  PQ/PR = NQ/QR .....................(2)

From Δ PQR

PQ2 + QR2 = PR2

4A2/ b2 + b2 = PR2

1649_PR.png

Equation (2) becomes

2 A/bxPR  = NQ/b

NQ = 2A/PR

NQ = 2 A/√4 A2 + b4


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