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.

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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

Equation (2) becomes

2 A/bxPR = NQ/b

NQ = 2A/PR

NQ = 2 A/√4 A² + b⁴

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