Question

if the diagonals of a quadrilateral are equal and bisect each other at right angle then prove that it is a square

diagram also needed and solve it in paper and send !!


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Answer 1

$\huge \mathscr{\orange {\underline{\pink{\underline {✿Answer✿:-}}}}}$

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= 900    ..........(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= 1800

∠B+∠B= 1800

∠B= 900

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

Answer 2

Answer:

nsuwer:-

Given,

Diagonals are equal

AC-BD

(1)

and the diagonals bisect each other at

right angles

OA=OC:B=OD

. (2)

LAOB 4BOC= 4COD= LAOD= 900

*****.5)

Proof.