Mathematics
In a quadrilateral ABCD, ∠B = 90° = ∠D. Prove that :
2AC2 - BC2 = AB2 + AD2 + DC2.
Pythagoras Theorem
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Answer
Given: In a quadrilateral ABCD, ∠B = 90° = ∠D.
To prove: 2AC2 - BC2 = AB2 + AD2 + DC2
Construction: Join diagonal AC.
Proof : Δ ABC and Δ ADC are two right angled triangles.
In Δ ABC, using Pythagoras theorem,
AC2 = AB2 + BC2 ……………..(1)
In Δ ADC, using Pythagoras theorem,
AC2 = AD2 + DC2 ……………..(2)
Adding (1) and (2), we get:
⇒ AC2 + AC2 = AB2 + BC2 + AD2 + DC2
⇒ 2AC2 = AB2 + BC2 + AD2 + DC2
⇒ 2AC2 - BC2 = AB2 + AD2 + DC2
Hence, 2AC2 - BC2 = AB2 + AD2 + DC2.
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