Question

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

I will surely mark as brainliest.

15 points please answer fast

you can also send a photo

Answer 1

Let ABCD be quadrilateral
Then diagonals AC and BD and let them meet at O
Now since diagonals are equal and bisect each other at right angle
Therefore AO=OC= BO=OD
IN TRIANGLE ABO and ADO AO=OD
ANGLE AOB=ANGLE ADO {EACH90}°
AND OA =OA
Therefore AB =AD. (CPCT)
Similarly in other triangles AOD,DOC AND DOC,BOC
AD=DC AND DC= BC
WE GET
AB=BC =CD=AD
NOW IN TRIANGLE BAD AND TRIANGLE ABC
AD=BC. PROVED ABOVE
BD=AC. PROVED ABOVE
AND BA=AB. Common
TRIANGLE BAD CONGRUENT TRIANGLE ABC BY SSS CONGRUENCE
THEREFORE BAD=ABC
SIMILARLY IN TRIANGLE ABC ,TRIANGLE BCD AND TRIANGLE BCD AND ADC
WE PROVE
ANGLE ABC=ANGLE BAD=angle ADC=angle BCD
SUM OF ANGKES OF A QUADRILATERAL IS 360°
THEREFORE SUMOF ALL 4 ANGLES IS 360°
Now all angles are equal
Therefore 4 ABC=360
ANGLE ABC=90°
THEREFORE SINCE ALL ANGLES ARE EQUAL
ANGLE ABC=ANGLE BCD=ANGLE BAD =ANGLE ADC.
NOW quadrilateral WITH ALL SIDES EQUAL AND EACH ANGLE 90° IS SQUARE