# If diagonals of a quadrilateral are equal and bisect each other at right angle then it is square

1
by sam34

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by sam34

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TO PROVE - AC = BD,

OA=OC, OB =OD

∠AOB =90°

PROOF -

1) IN ΔABC AND ΔDCB

AB=DC ( EQUAL SIDES OF SQUARE)

∠ABC=∠DCB (90° EACH)

BC = CB(COMMON SIDE)

∴ΔABC ≡ ΔDCB (SAS RULE)

AC=DB ( CPCT)

2) IN ΔAOB AND ΔCOD

∠AOB=∠COD (VERTICALLY OPPOSITE ANGLES )

∠ABO=∠CDO (ALTERNATE INTERIOR ANGLES )

AB = CD ( EQUAL SIDES OF SQUARE )

ΔAOB ≡ ΔCOD ( AAS RULE )

AO=CO AND OB = OD ( CPCT)

3) IN ΔAOB AND ΔCOB

AO = CO(PROVED ABOVE)

AB=CB ( EQUAL SIDES OF SQUARE)

BO=BO (COMMON)

ΔAOB≡ΔCOB ( SSS RULE)

∴∠AOB=∠COB ( CPCT )

∠AOB+∠COB=180° ( LINEAR PAIR )

2∠AOB=180° (SINCE,∠AOB=∠COB)

∠AOB=180°/2

∠AOB=90°

THEREFORE, PROVED THAT IN A QUAD. IF DIAGONALS ARE EQUAL AND THEY PERPENDICULARLY BISECT EACH OTHER THEN THEY THE QUAD. IS A SQUARE.