Question

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

Answer 1

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$\huge{\underline{\underline{\sf{\orange{</p><p>Question:-}}}}}$

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

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$\huge{\underline{\underline{\sf{\orange{</p><p>Solution:-}}}}}$

${\blue{\sf\underline{Given}}}$

A quad. ABCD in which the diagonals AC and BD intersect at O such that AC = BD;OA and OB = OD; AC ⊥ BD.

${\blue{\sf\underline{To\:prove}}}$

ABCD is a square

${\blue{\sf\underline{Proof}}}$

Since the diagonals of quad. ABCD bisect each other, therefore ABCD is a ||gm.

Now,in ∆ABO AND ADO, we have

OB = OD (given)

OA = OA (common)

∠AOB = ∠AOD = 90° [∴AC ⊥ BD]

∴∆ABO ≅ ∆ADO (SAS-criterion).

And so, AB = AD (c.p.c.t).

Now,AB = CD and AD = BC [∴opp. sides of a ||gm are equal]

∴AB = BC = CD = AD.

Again, in ∆ABC and BAD,we have

AB = BA (common)

AC = BD (given)

BC = AD (proved)

∴ ∆ABC ≅ ∆BAD (SSS-criterion).

And so, ∠ABC = ∠BAD (c.p.c.t)

But,∠ABC + ∠BAD = 180°

∴∠ABC = ∠BAD =90°

Thus,AB = BC = CD = AD and ∠A = 90°.

∴ABCD is a square.

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Answer 2

Answer:

Toprove

ABCD is a square

Prove

Since the diagonals of quad. ABCD bisect each other, therefore ABCD is a ||gm.

Now,in ∆ABO AND ADO, we have

OB = OD (given)

OA = OA (common)

∠AOB = ∠AOD = 90° [∴AC ⊥ BD]

∴∆ABO ≅ ∆ADO (SAS-criterion).

And so, AB = AD (c.p.c.t).

Now,AB = CD and AD = BC [∴opp. sides of a ||gm are equal]

∴AB = BC = CD = AD.

Again, in ∆ABC and BAD,we have

AB = BA (common)

AC = BD (given)

BC = AD (proved)

∴ ∆ABC ≅ ∆BAD (SSS-criterion).

And so, ∠ABC = ∠BAD (c.p.c.t)

But,∠ABC + ∠BAD = 180°

∴∠ABC = ∠BAD =90°

Thus,AB = BC = CD = AD and ∠A = 90°.

∴ABCD is a square.