Question

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

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)

∠AOB= ∠BOC= ∠COD= ∠AOD= 90

0

(3)

Proof:

Consider △AOB and △COB

OA=OC...[from (2)]

∠AOB= ∠COB

OB is the common side

Therefore,

△AOB≅ △COB

From SAS criteria, AB=CB

Similarly, we prove

△AOB≅ △DOA, so AB=AD

△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 △ABC and △DCB

AC=BD (from (1))

AB=DC (from (4)

BC is the common side

△ABC≅ △DCB

So, from SSS criteria, ∠ABC= ∠DCB

Now,

AB∥CD, BC is the transversal

∠B+∠C= 180

0

∠B+∠B= 180

0

∠B= 90

0

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

0

So, ABCD is a square.

Hence proved.