wiz-icon
MyQuestionIcon
MyQuestionIcon
8
You visited us 8 times! Enjoying our articles? Unlock Full Access!
Question

Question 7
AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters; (ii) ABCD is a rectangle.

Open in App
Solution


Let two chords AB and CD are intersecting each other at point O

In ΔAOBandΔCOD,
OA = OC (Given)
OB = OD (Given)
AOB=COD (Vertically opposite angles)
ΔAOBΔCOD (SAS congruence rule)
AB = CD (By CPCT)
Similarly, it can be proved that ΔAODΔCOB
and AD = CB (By CPCT)
Since in quadrilateral ABCD, opposite sides are equal in length, ABCD is a parallelogram.
We know that opposite angles of a parallelogram are equal.
A=C
However, A+C=180 (ABCD is a cyclic quadrilateral)
A+A=180
2A=180
i,e A=90
As ABCD is a parallelogram and one of its interior angles is 90, therefore, it is a rectangle.
A is the angle subtended by chord BD and A=90, therefore, BD should be the diameter of the circle. Similarly, AC is the diameter of the circle.






flag
Suggest Corrections
thumbs-up
1
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Circles and Quadrilaterals - Theorem 11
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon