Topic : Quadrilateral
Question : If in a quadrilateral ABCD, the diagonals are perpendicular bisectors of each other, then prove that ABCD is a square.
Solution :
Given : ABCD is a quadrilateral
AC = BD
AC ┴ BD
AO = OC , BO = OD
To prove : ABCD is a square
Proof : In ∆AOB and ∆BOC
AO = OC (given)
BO = BO (common)
∟BOA = ∟BOC = 90º (given)
So by SAS theorem of congruency
∆ADB congruent to ∆BDC
Therefore AB = BC (corresponding sides of congruent triangles)
Similarly we can prove that
BC = CD , CD = DA , DA = AB
∆DAB congruent to ∆BAC
Therefore ∟DAB = ∟CBA
So all the sides are equal and each angle = 90º
Hence proved ABCD is a square
