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
Also AD || BC ...............2
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.