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

Asked by Kalai(KENDRIYA VIDYALAYA) , on 5/12/12

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 âˆ AOB = âˆ BOC = âˆ COD = âˆ AOD = 90Âº.

Now, to prove ABCD is a square, we have to prove that ABCD is a parallelogram, AB = BC = CD = AD, and one of its interior angles is 90Âº.

Proof:

In Î”AOB and Î”COD,

AO = CO (Diagonals bisect each other)

OB = OD (Diagonals bisect each other)

âˆ AOB = âˆ COD (Vertically opposite angles)

âˆ´ Î”AOB â‰… Î”COD (SAS congruence rule)

âˆ´ AB = CD (By CPCT) ... (1)

And, âˆ OAB = âˆ 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.

âˆ´ AB || CD ... (2)

From equations (1) and (2), we obtain

ABCD is a parallelogram.

In Î”AOD and Î”COD,

AO = CO (Diagonals bisect each other)

âˆ AOD = âˆ COD (Given that each is 90Âº)

OD = OD (Common)

âˆ´ Î”AOD â‰… Î”COD (SAS congruence rule)

âˆ´ AD = DC ... (3)

However, AD = BC and AB = CD (Opposite sides of parallelogram ABCD)

âˆ´ AB = BC = CD = DA

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

AC = BD (Given)

DC = CD (Common)

âˆ´ Î”ADC â‰… Î”BCD (SSS Congruence rule)

âˆ´ âˆ ADC = âˆ BCD (By CPCT)

However, âˆ ADC + âˆ BCD = 180Â° (Co-interior angles)

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 90Âº. Therefore, ABCD is a square.

Posted by Ankush Jainon 5/12/12

This conversation is already closed by Expert

I WILL SOON GVE U DIS ANS

Posted by Sanika(CM Academyankleshwargujarat) on 9/10/11

can u wait?

Posted by Sanika(CM Academyankleshwargujarat) on 9/10/11

Using the same figure,

If DO=AO

(Angles opposite to equal sides are equal)

So, all angles of the quadrilateral are right angles making it a square.

6. Diagonal AC of a parallelogram ABCD bisects angle A . Show that

(i) it bisects angle C also,

(ii) ABCD is a rhombus.

Answer: ABCD is a parallelogram where diagonal AC bisects angle DAB

As diagonals are intersecting at right angles so it is a rhombus

7. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that:

With equal opposite angles and equal opposite sides it is proved that APCQ is a parallelogram

Posted by Sanika(CM Academyankleshwargujarat) on 9/10/11

sorry sorry....!!dis was wrng

Posted by Sanika(CM Academyankleshwargujarat) on 9/10/11

If DO=AO

1. angle DAO =angle BAO=4degree

(Angles opposite to equal   sides are equal)

So, all angles of the quadrilateral are right angles making it a square

Posted by Sanika(CM Academyankleshwargujarat) on 9/10/11

here,u make a sqyare ABCD and make its diagnols which intersect at o.

Posted by Sanika(CM Academyankleshwargujarat) on 9/10/11

It has a simple solution.

Diagonals are equal only in sqaure and rhombus

And

Diagonals bisect each other at right angle only in square and rectangle

So, Square is the only figure in which diagonals are equal and bisect each other at right angle

Posted by Payal Aggarwal(VIDYA NIKETAN HIGH SCHOOL NO.2) on 9/10/11

in //gm ABCD, AC and BD are diagonals of parallelogram

PROVE:OA=OC & OB=OD

ABCD is a parallelogram ,therefore

in triangle AOB and COD

AB=CD(opp sides of parallelogram are equal)

therefore triangle AOBtriangleCOD(ASA congurence criterian)

OA=OC & OB=OD(cpct)

hence, prove that diangonals of parallelogram bisect each other. in //gm ABCD, AC and BD are diagonals of parallelogram

PROVE:OA=OC & OB=OD

ABCD is a parallelogram ,therefore

in triangle AOB and COD

AB=CD(opp sides of parallelogram are equal)

therefore triangle AOBtriangleCOD(ASA congurence criterian)

OA=OC & OB=OD(cpct)

hence, prove that diangonals of parallelogram bisect each other.

Posted by Param Sukhadia(Kendriya Vidhyalaya No. 1) on 17/11/12

in //gm ABCD, AC and BD are diagonals of parallelogram

PROVE:OA=OC & OB=OD

ABCD is a parallelogram ,therefore

in triangle AOB and COD

AB=CD(opp sides of parallelogram are equal)

therefore triangle AOBtriangleCOD(ASA congurence criterian)

OA=OC & OB=OD(cpct)

hence, prove that diangonals of parallelogram bisect each other.

Posted by Param Sukhadia(Kendriya Vidhyalaya No. 1) on 17/11/12

shwetha i can feel your tits from here

Posted by U Suckon 4/12/12