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AB = BC and AD is perpendicular to CD. Prove that AC2 = 2.BC. DC

Question: In the figure AB = BC and AD is perpendicular to CD. Prove that AC2 = 2.BC. DC

solution : In ∆ ADC
AC2 = AD2 + DC2
AC2 = AB2 - BD2 + DC2(AS ABD is a right angled triangle)
or AC2 = AB2 + DC2 - BD2
AC2 = AB2 + DC2 - ( DC-BC )2
or AC2 = AB2 + DC2 - (DC2 + BC 2 -2 DC. BC)
or AC2 = AB2 + DC2 - DC2 - BC 2 + 2 DC. BC
or AC2 = AB2 - AB 2 + 2 DC. BC (As AB = BC )
.'. AC2 = 2 DC. BC Proved

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