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Question

AC and BD are chords of a circle which bisect each other. Prove that ABCD is a rectangle.

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Solution

AO=CO ....AC and BD bisect each other
BO=DO ....AC and BD bisect each other
ABO=COD ...Vertically opposite angles
AOBCOD ...[SAS test]
BAO i.e., BAC=ABO i.e., ABD
ACBD
Again, AODCOB ...[SAS test]
ADCB
ABCD is a cyclic parallelogram.
DAC=DBA.............. (1)
[Opp.angles of a parallelogram are equal]
Also, ACBD is a cyclic quadrilateral
Therefore DAC+DBA=180
................ (2)
From (1) and (2), we get
DAC=DBA=90
Hence, ABCD is a rectangle.

496306_464064_ans.png

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