Circles-HOTS Questions

1.   In the figure, OD is perpendicular to chord AB of a circle whose centre is O. If BC is a diameter; prove that CA = 2OD.

2.   l is a line intersecting two concentric circles having common centre O, at A, B, C and D. Prove that AB = CD.

3.   AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Prove that EB = ED.

4.   If O be the centre of the circle, find the value of �x� in each of the following figures.

5.   Prove that equal chords of a circle subtend equal angles at the centre.

6.   The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. Prove it.

7.   Prove that equal chords of a circle (or congruent circles) are equidistant from the centre (or centres).

8.   In the figure, OD is perpendicular to the chord AB of a circle with centre O. If BC is a diameter, show that AC || OD and AC = 20D.

           Hint: ∴ OD ⊥ AB               therefore; D is the mid-point of AB.

9.   If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.

10.   Show that the angles in the same segment of a circle are equal.