Question

5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right
angles, then it is a square​

Answer 1

Step-by-step explanation:

GIVEN : Diagonals are equal

AC=BD —( 1 )

and the diagonals bisect each other at right angles

OA = OC ; OB = OD — ( 2)

angle AOB = Angle BOC = angle COD = angle AOD = 90 degree — ( 3 )

PROOF :

Consider Triangle AOB and triangle COB

OA = OC — [ from(2)]

angle AOB = angle COB

OB is the common side

Therefore,

triangle AOB ≈ triangle COB

From SAS, AB = CB

Similarly we prove,

triangle AOB ≈ triangle DOA, so AB = AD

triangle BOC ≈ COD, so CB = DC

So, AB = AD = CB = DC —( 4 )

So, in quadrilateral ABCD,both pairs of opposite sides are equal hence ABCD is a parallelogram

in triangle ABC and triangle DCB

AC = BD — ( from 1 )

AB = DC — ( from 4 )

BC is the common side

triangle ACB is congruent to triangle DCB

NOW,

AB ll CD, BC is the transversal

Angle B + angle C = 180 degree

Angle B + Angle B = 180 degree

Angle B = 90°

Hence, ABCD is a parallelogram with all sides equal and one angle is 90 degree

so, ABCD is a square

HENCE PROVED