Q. 224.5( 4 Votes )

In Δ ABC, ∠ B = 90° and D is the mid-point of BC. Prove that AC2 = AD2 + 3CD2.

Answer :

Given: In Δ ABC, ∠B = 90° and D is the mid-point of BC.
To Prove: AC2 = AD2 + 3CD2
Proof:
In Δ ABD,
AD2 = AB2 + BD2
AB2 = AD2 - BD2 ……………..(i)
In Δ ABC,
AC2 = AB2 + BC2
AB2 = AC2 - BC2 ……………..(ii)
Equating (i) and (ii)
AD2 - BD2 = AC2 - BC2
AD2 - BD2 = AC2 – (BD + DC)2
AD2 - BD2 = AC2 – BD2 - DC2 – 2BD × DC
AD2 = AC2 - DC2 – 2DC2 (DC = BD)
AD2 = AC2 - 3DC2

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