In Fig. 6.31, OA . OB = OC . OD. Show that A = C and B = D.
Step-by-Step Solution
Key Concept: The given relation OA·OB = OC·OD is the condition for the four points A, B, C, D to be concyclic (Power of a Point theorem). Once the quadrilateral ABCD is cyclic, equal angles subtend equal chords: ∠A and ∠C subtend the same chord BD, and ∠B and ∠D subtend the same chord AC.
1. Given: OA·OB = OC·OD, where O is an interior point and OA, OB, OC, OD are straight lines meeting the circle at A, B, C, D respectively.\
2. Power of a Point: For a point O inside a circle, if two chords AB and CD intersect at O, then OA·OB = OC·OD. The converse is also true – if OA·OB = OC·OD, the four points A, B, C, D lie on a single circle. Hence, A, B, C, D are concyclic.\
3. Cyclic Quadrilateral Property: In a cyclic quadrilateral, equal chords subtend equal angles at the circumference.\
4. Identify the chords: In the cyclic quadrilateral ABCD, chord BD subtends ∠A at point A and ∠C at point C. Similarly, chord AC subtends ∠B at point B and ∠D at point D.\
5. Apply the property: Since the same chord BD subtends ∠A and ∠C, we have \(\angle A = \angle C\). Likewise, the same chord AC subtends \(\angle B\) and \(\angle D\), giving \(\angle B = \angle D\).\
6. Conclusion: Hence, \(\angle A = \angle C\) and \(\angle B = \angle D\).
Correct Answer: ∠A = ∠C and ∠B = ∠D