Question

if ABCD is a quadrilateral such that AB=AD and CB= CD, then prove that AC is the perpendicular bisect of BD

Answer 1

In the given image above.
AB = AD
and
DC = BC

Now,
in traingle ACD and ABC

AD= AB (given)
DC = BC (given)
AC = AC (common)

Therefore,
Triangle ADC is congruent to triangle ABC By SSA congurence creteria.

Therefore,

By C.P.C.T.

angle DAC = angle BAC
Now in Triangle AOD and triangle AOB

angle DAC = angle BAC (proved above)
AD = AB (given)
AO = AO (common)
Therefore,
both the given triangles are congruent by SAS creteria.

now,

By CPCT
DO = OB
Angle AOD = angle AOB

by using linear pair,
angle AOD + angle AOB = 180°

(as both the angles are equal )

2 angle AOD = 180°

angle AOD = 90°

Similarly,
angle AOB = 90°.

Using the same technique,

In triangle DOC and BOC.

AO= OC

angle DOC = angle BOC = 90°



Hence, proved.

Answer 2

Answer:

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