Mathematics
In a quadrilateral ABCD, ∠B = 90° and ∠D = 90°. Prove that :
2AC2 - AB2 = BC2 + CD2 + DA2.
Pythagoras Theorem
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Answer
In right-angled triangle ABC,
By pythagoras theorem,
⇒ AC2 = AB2 + BC2
⇒ AB2 = AC2 - BC2 ………(1)
In right-angled triangle ADC,
By pythagoras theorem,
⇒ AC2 = AD2 + DC2 ……..(2)
To prove :
2AC2 - AB2 = BC2 + CD2 + DA2.
Solving L.H.S. of the equation :
⇒ 2AC2 - AB2 = 2AC2 - (AC2 - BC2) [From equation (1)]
= 2AC2 - AC2 + BC2
= AC2 + BC2
= AD2 + DC2 + BC2 [From equation (2)]
= R.H.S.
Hence, proved that 2AC2 - AB2 = BC2 + CD2 + DA2.
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