AC and BD are chords of circle that bisect each other. Prove that: (i) AC and BD are diameters. (ii) ABCD is a rectangle.
Sol. (i)Suppose the two chords AC and BD bisect each other at O. and are joined.
Now, in ΔAOD and ΔCOB,
[Vertically opposite angles]
[By SAS congruence]
Similarly, chord chord .
Adding (i) and (ii) we get
∴ BD divides the whole circle into two semi-circles.
⇒ BD is the diameter of circle.
Hence, AC is also the diameter of circle.
(ii) Since, AC and BD are the diameters of a circle
Now, in quadrilateral
and diagonal diagonal
∴ ABCD is a rectangle.