# Q: 7    AC and BD are chords of a circle which bisect each other. Prove that           (i) AC and BD are diameters

Given: AC and BD are chords of a circle which bisect each other.

To prove: AC and BD are diameters.

Construction : Join AB,BC,CD,DA.

Proof :

In  ABD and CDO,

AO = OC     (Given )

AOB = COD  (Vertically opposite angles )

BO = DO        (Given )

So,   ABD CDO     (By SAS)

BAO = DCO    (CPCT)

BAO and DCO are alternate angle and are equal .

So,     AB || DC ..............1

From 1 and 2,

......................3(sum of opposite angles)

A = C    ................................4(Opposite angles of the parallelogram )

From 3 and 4,

BD is a diameter of the circle.

Similarly, AC is a diameter.

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