Question

Abcd is a quadrilateral that ab=ad and cb=cd.Prove that ac is perpendicular bisector of bd

Answer 1



Answer : As given , let ABCD be a quadrilateral wherein AB=AD and CB=CD.

To prove :  AC is the perpendicular bisector of BD

 

Consider triangle ADB , as AB=AD, it is an isosceles triangle.

=> by property of isosceles triangle , angle ADB = angle ABD 

 

therefore, triangle ADB is similar to triangle ABD.

 

Now of similar triangle => side OD = side OB.

  => AB/OB = AD/OD

=> AO is a bisector of BD.

 

 

similarly, in triangle BCD, BC= CD

=> it is also isosceles triangle , therefore angle CDB=angle CBD.

and hence , triangle CDB is similar to triangle CBD.

=> side OB= side OD.

=> CD/OD  = CB/OB

=> CO is a bisector of BD.

 

As OA and OC is the bisector of triangle ABD and triangle BCD respectively. Therfore AC is a bisector of BD and 

perpendicular to BD.

Hence proved.

Answer 2

in quadrilateral ABCD
ac=ad and cb= cd
here adjacent sides are equal. So it is a kite.
In kite the longer diagonal bisect the shorter diagonal at 90°. if bd is shorter diagonal than the AC will bisect it at 90°
Hence Proved
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