Math, asked by sharmaanjali69898, 7 months ago

5. Show that if the diagonals of a quadrilateral are equal and bisect each other a
angles, then it is a square.
Dr
99
n

Answers

Answered by helenfaustina12

Answer:

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

..........(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 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 tansversal

∠B+∠C= 180

∠B+∠B= 180

∠B= 90

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

So, ABCD is a square.

Hence proved.

solution

Answered By

toppr

8870 Views

How satisfied are you with the answer?

This will help us to improve better

answr

Get Instant Solutions, 24x7

No Signup required

girl

5. Show that if the diagonals of a quadrilateral are equal and bisect each other aangles, then it is a square.Dr99n​

Answers

5. Show that if the diagonals of a quadrilateral are equal and bisect each other a
angles, then it is a square.
Dr
99
n