Question

Abcd is a quadrilateral such that ab=ad and cb=cd prove that ac is the perpendicular bisector of bd

Answer 1

hey there ,

=) here ,given abcd is a quadrilateral such that ab = ad and cb = cd

◆ According to the property of quadrilateral the quadrilateral whose adjacent sides are equal then it is rhombus.

=) And we know tha diagonal ac and bd of rhombus are perpendicular bisector.

Hence, it is proved
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Answer 2

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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