CBSE Important Questions

Circles

Section A

Prove that the following:

1. Equal chord of a circle subtend equal angles at the centre.
2. The perpendicular from the centre of a circle to a chord bisects the chord.
3. The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
4. Chords equidistant from the centre of a circle are equal in length.
5. The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
6. Angles in the same segments of a circle are equal.
7. The sum of either pair of opposite angles of a cycle quadrilateral is 180⁰.
8. If the sum of a pair of opposite angles of a quadrilateral is 180⁰, the quadrilateral is cyclic.

Section B

1. 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.
2. AB is a diameter of the circle, CD is chord equal to the radius of the circle. AC and BD when extended intersect at a point E. Prove that AEB-60⁰

   

3. ABCD is a cycle quadrilateral in which AC and BD are its diagonals. If DBC=55⁰ and BAC =45⁰, find BCD.

   

4.   Two circles intersect at two points A and B. AD and AC are diameter to the two circles. Prove that B lies on the line segment DC.
5. Prove that the quadrilateral formed by the internal angle bisectors of any quadrilateral is cyclic.
6. Prove that if chord of congruent circles subtend equal angles at their centers, then the chords are equal.
7. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4cm. Find the length of the common chord.
8. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
9. If two equal chords of a circle intersect within the circle. Prove that the line joining the point of intersection to the centre makes equal angles with the chords.
10. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.
11. ABCD is a cyclic quadrilateral whose diagonal intersect at a point E. If DBC=70⁰, BAC is 30⁰, find BCD. Further, if AB =BC, find ECD.
12. If a diagonal of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
13. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
14. If circle are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
15. ABC and ADC are two right triangles with common hypotenuse AC. Prove that CAD= CBD.
16. Prove that a cyclic parallelogram is a rectangle