1. “Two equal circles are such that the center of one lies on the other, and they intersect at points A and B. A straight line through A intersects the circles again at points C and D. Prove that triangle BCD is equilateral.”

2. Two identical circles, each with radius \(r\), intersect in such a way that each circle passes through the center of the other. The centers of the circles are labeled A and B, and they intersect at points P and Q. The area of triangle \(\triangle APB\) will be:

(a) \(\cfrac{\sqrt3}{4}r^2\) (b) \(\cfrac{\sqrt3}{2}r^2\) (c) \(\cfrac{\sqrt3}{3}r^2\) (d) \(\sqrt3 r^2\)

3. The centers of two circles are P and Q; the circles intersect at points A and B. A line parallel to the line segment PQ is drawn through point A, and it intersects the two circles at points C and D. Prove that CD = 2PQ.

4. Two circles with centers P and Q intersect at points A and B. A straight line parallel to PQ passes through point A and intersects the two circles at points C and D respectively. If PQ = 5 cm, find the length of CD.

5. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

6. I have drawn two circles with centers C and D that intersect each other at points A and B. A straight line PK is drawn through point A, which intersects the circle with center C at point P and the circle with center D at point Q. Prove that: (i) \(\angle PBQ = \angle CAD\) (ii) \(\angle BPC = \angle BQD\)

7. I have drawn two circles with centers A and B that externally touch each other at point O. A straight line is drawn through point O, which intersects the two circles at points P and Q respectively. Prove that AP is parallel to BQ.

8. In the adjacent figure, two circles intersect at points C and D. Two straight lines passing through points D and C intersect one circle at points A and B respectively, and the other circle at points E and F respectively. If ∠DAB = 75°, then find the value of ∠DEF.

(a) 75° (b) 70° (c) 60° (d) 105°

9. The centers of two circles are P and Q. The circles intersect at points A and B. Through point A, two straight lines parallel to the line segment PQ intersect the circles at points C and D. Prove that CD = 2PQ.

10. Two circles with centers X and Y intersect at points A and B. The midpoint S of the line segment XY is joined with point A, and from point A, a perpendicular is drawn on line SA, which intersects the two circles at points P and Q. Prove that PA = AQ.

11. We have drawn a median \(AD\) of triangle \( \triangle ABC \). If a straight line parallel to \(BC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that the line segment \(PQ\) is bisected by the median \(AD\).

12. I have drawn two circles that intersect each other at points G and H. Then I drew a straight line through point G which intersects the two circles at points P and Q. Next, I drew another straight line through point H, parallel to PQ, which intersects the two circles at points R and S. Prove that PQ = RS.

13. Two circles intersect at points A and B, and the center of one of the circles is point O. A straight line from point A intersects the circle centered at O at point R and the other circle at point P. By joining points P to B and R to B, prove that PR = PB.

14. In the adjacent figure, two circles with centers P and Q intersect at points B and C. ACD is a straight line segment. If ∠ARB = 150° and ∠BQD = x°, then find the value of x.

15. Two circles intersect at points A and B. Two straight lines passing through points A and B intersect the circles at points P, Q and R, S respectively. If \(\angle\)APR = 80° and \(\angle\)BRP = 78°, then find the values of \(\angle\)AQS and \(\angle\)BSQ.

16. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC?

(a) 1:2 (b) 2:1 (c) 1:1 (d) None of these

17. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?

(a) 145° (b) 55° (c) 110° (d) None of the above

18. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

19. Two circles intersect each other at points P and Q. If PA and PB are the diameters of the respective circles, then prove that the points A, Q, and B lie on a straight line.

20. Draw a median \(AD\) of \(\triangle ABC\). If a straight line parallel to \(BC\) intersects \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that \(AD\) bisects the line segment \(PQ\).

21. The centers of two concentric circles are at point O. A straight line intersects one circle at points A and B, and the other circle at points C and D. If AC = 5 cm, find the length of BD.

(a) 2.5 cm (b) 5 cm (c) 10 cm (d) none of the above

22. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

23. In triangle ∆ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that PB = AQ, AP = 9 units, and QC = 16 units, what is the length of PB?

(a) 12 cm (b) 6 cm (c) 8 cm (d) 10 cm

24. In a circle with center O, PQ and PR are two chords. Tangents drawn at points Q and P intersect at point S. If ∠QSR = 70°, then what is the measure of ∠QPR?

25. A straight line intersects one of two concentric circles at points A and B, and the other at points C and D. Prove that AC = BD.

26. Two tangents are drawn to a circle from points A and B on its circumference, and they intersect at point P. If \(\angle\)APB = 68°, then what is the measure of \(\angle\)PAB?

27. In \(\triangle\)ABC, a line parallel to side BC intersects AB and AC at points P and Q, respectively. If AP = 4 cm, QC = 9 cm, and PB = AQ, then what is the length of PB?

28. Sahana has drawn two circles that intersect at points P and Q. If PA and PB are the diameters of the respective circles, prove that the points A, Q, and B lie on a straight line.

29. Rajat has drawn a straight line segment PQ whose midpoint is R. He then drew two circles using PR and PQ as diameters. I drew a straight line passing through point P that intersects the first circle at point S and the second circle at point T. Prove with reasoning that PS = ST.

30. In the adjacent figure, a circle is centered at point O. From an external point C, two tangents are drawn to the circle, touching it at points P and Q respectively. Another tangent is drawn at point R on the circle, which intersects the tangents CP and CQ at points A and B respectively. If CP = 11 cm and BC = 7 cm, then find the length of BR.