1. A straight line intersects two concentric circles at points A and B on one circle and at points C and D on the other. Prove with reasoning that AC = DB.

2. 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

3. 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.

4. “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.”

5. Timir has drawn two circles that intersect each other at points P and Q. From point P, two straight lines are drawn: one intersects one circle at points A and B, and the other intersects the other circle at points C and D respectively. Prove that \(\angle AQC = \angle BQD\).

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 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.

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. Two circles intersect at points A and B, and the circumference of each passes through the center of the other. If a straight line drawn through point A intersects the two circles again at points C and D respectively, then what type of triangle is BCD?

(a) right-angled isosceles (b) scalene (c) isosceles (d) equilateral

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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.

16. Two circles intersect each other at points P and Q. Two straight lines passing through points P and Q intersect one circle at points A and C, and the other circle at points B and D respectively. Prove that AC is parallel to BD.

17. Two concentric circles have radii of 13 cm and 15 cm, respectively. A chord AB of the larger circle intersects the smaller circle at points P and Q. If PQ = 10 cm, then AB will be:

(a) 28 cm (b) 20 cm (c) 18 cm (d) 16 cm

18. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP : PB = 2 : 1 \) and \( AC = 18 \) cm, find the length of \( AQ \).

(a) 12 cm (b) 9 cm (c) 6 cm (d) None of the above

19. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP = 18 \) cm, \( QC = 9 \) cm, and \( AQ = 2 \times PB \), find the length of \( PB \).

(a) 6 cm (b) 12 cm (c) 18 cm (d) 9 cm

20. 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

21. In triangle ∆ABC, a straight line parallel to side BC intersects AB and AC at points P and Q respectively. Given that \(\frac{AQ}{QC} = \frac{3}{4}\) and AB = 21 cm, find the length of PB.

22. If AB and AC are chords of the larger of two concentric circles, and they touch the smaller circle at points P and Q respectively, prove that: \[ PQ = \frac{1}{2}BC \]

23. In two concentric circles, the chords AB and AC of the larger circle touch the smaller circle at points P and Q respectively. Prove that PQ = \(\frac{1}{2}\)BC.

24. 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.

25. In two concentric circles, the chords AB and AC of the larger circle touch the smaller circle at points P and Q respectively. Prove that \(PQ = \frac{1}{2}BC\).

26. 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.

27. 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.

28. 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

29. In triangle ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. AP = QC, AB = 12 cm, AQ = 2 cm. Find the length of CQ.

(a) 4 cm (b) 6 cm (c) 9 cm (d) None of the above

30. 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