Question

ABCD is a quadrilateral such that AB=AD and CB=CD prove that AC is the perpendicular bisector of BD

Answer 1

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Answer 2

Answer:

Here's your solution ✌️

Given- AB=AD

DC=BC

To prove- AC is the perpendicular bisector of BD

Construction- Join AC and BD

Proof- In ∆ ADC and ∆ ABC

AD = AB. (Given)

DC = BC. (Given)

AC = AC. ( Common)

=>∆ ADC and ∆ ABC are congruent by SSS rule

Angle DAC = Angle BAC ( CPCT)

In ∆ AOD and ∆ AOB

AB = AD. (Given)

Angle DAC = Angle BAC. (Proved)

AO = AO. (Common)

=>.∆ AOD and ∆ AOB are congruent by SAS rule DO = BO. ( CPCT)

Angle AOD = Angle AOB (CPCT)

But,

Angle AOD + Angel AOB = 180°.

(Linear pair)

2 (Angle AOD) = 180°

Angle AOD = 180°/ 2

Angle AOD = Angle AOB = 90°

Similarly,

Angle DOC = Angle BOC = 90°

=> AC is the perpendicular bisector of BD

Hence Proved.

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