Q. 53.8( 435 Votes )

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

Answer :

Let us consider a quadrilateral ABCD in which the diagonals AC and BD intersect each other at O.

It is given that the diagonals of ABCD are equal and bisect each other at right angles.

Therefore, AC = BD, OA = OC, OB = OD, and

AOB = BOC = COD = AOD = 900



To prove: ABCD is a square,
Proof:

we have to prove that ABCD is a parallelogram with all of its sides equal and one of the interior angle is right angle.

or,
AB = BC = CD = AD, and one of its interior angles is 900

In ΔAOB and ΔCOD,

AO = CO (Diagonals bisect each other)

OB = OD (Diagonals bisect each other)

AOB = ∠ COD (Vertically opposite angles)

(SAS) Definition: Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles. 

ΔAOB ΔCOD (SAS congruence rule)

AB = CD (By CPCT)  ......... eq(1)

And,

OAB = OCD (By CPCT)

However, these are alternate interior angles for line AB and CD and alternate interior angles are equal to each other only when the two lines are parallel

AB || CD  ............................eq(2)

From equations (1) and (2), we obtain ABCD is a parallelogram 
          
(Theorem: Quadrilateral with any of opposite sides equal and parallel is a parallelogram.

In ΔAOD and ΔCOD,

AO = CO (Diagonals bisect each other)

∠ AOD = ∠ COD (Given that each is 900)

OD = OD (Common)

ΔAOD ΔCOD (SAS congruence rule)

AD = DC (3)

However,

AD = BC and

AB = CD (Opposite sides of parallelogram ABCD)

AB = BC = CD = DA

Therefore, all the sides of quadrilateral ABCD are equal to each other.


In ΔADC and ΔBCD,


AD = BC (Already proved)


AC = BD (Given)


DC = CD (Common)


ΔADC ΔBCD (SSS Congruence rule)


ADC = BCD (By CPCT)


However,


ADC + BCD = 1800 (Co-interior angles)


ADC + ADC = 1800


2 ADC = 1800


ADC = 900 One of the interior angles of quadrilateral ABCD is a right angle.


Thus, we have obtained that ABCD is a parallelogram, AB = BC = CD = AD and one of its interior angles is 900


Therefore, ABCD is a square

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Critical Thinking Problems on QuadrilateralsCritical Thinking Problems on QuadrilateralsCritical Thinking Problems on Quadrilaterals44 mins
Quiz | Properties of ParallelogramQuiz | Properties of ParallelogramQuiz | Properties of Parallelogram31 mins
Extras on QuadrilateralsExtras on QuadrilateralsExtras on Quadrilaterals40 mins
Smart Revision | QuadrilateralsSmart Revision | QuadrilateralsSmart Revision | Quadrilaterals43 mins
Quiz | Basics of QuadrilateralsQuiz | Basics of QuadrilateralsQuiz | Basics of Quadrilaterals36 mins
RD Sharma |  Extra Qs. of Cyclic QuadrilateralsRD Sharma |  Extra Qs. of Cyclic QuadrilateralsRD Sharma | Extra Qs. of Cyclic Quadrilaterals31 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses