Question
AC and BD are chords of circle that bisect each other. Prove that: (i) AC and BD are diameters. (ii) ABCD is a rectangle.
Solution
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Sol. (i)Suppose the two chords AC and BD bisect each other at O.  and  are joined.
Now, in ΔAOD and ΔCOB,
 [Given]
 [Given]
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[Vertically opposite angles]
[By SAS congruence]
[cpct]
⇒ 
…(i)
Similarly, chord 
chord 
.
⇒ 
…(ii)
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
and diagonal 
diagonal 
∴ ABCD is a rectangle.
