1. Two circles touch each other externally at point C. AB is a common tangent to both circles, touching them at points A and B respectively. Find the measure of ∠ACB.

(a) 60° (b) 45° (c) 30° (d) 90°

2. Two circles touch each other externally at point \(C\). \(AB\) is a common tangent to the two circles, touching them at points \(A\) and \(B\), respectively. The measurement of \(\angle ACB\) is –

(a) 60° (b) 45° (c) 30° (d) 90°

3. Two circles touch externally at point \(C\). A common external tangent to both circles touches them at points \(A\) and \(B\) respectively. The measure of \(\angle ACB\) is -

(a) 60° (b) 45° (c) 90° (d) 30°

4. Two circles centered at P and Q externally touch each other at point A. A common external tangent to both circles touches them at points R and S, respectively. Prove that \(\angle\)RAS = 90°.

5. Two circles touch each other externally at point C. A common tangent AB touches the circles at points A and B respectively. Find the value of \(\angle ACB\).

6. Two circles touch externally at point C. AB is a common tangent to both circles, touching them at points A and B. The measure of \(\angle\) ACB is?

(a) 60° (b) 45° (c) 90° (d) 75°

7. Two circles are externally tangent at point \( C \). \( AB \) is a common tangent to both circles, touching them at points \( A \) and \( B \). Determine the measure of \( \angle ACB \).

(a) 60° (b) 90° (c) 45° (d) 30°

8. Two circles touch each other externally at point \( C \). A common external tangent touches the two circles at points \( A \) and \( B \). Determine the value of \( \angle ACB \).

(a) 60° (b) 45° (c) 30° (d) 90°

9. We have drawn two circles with centers \(A\) and \(B\), which touch each other externally at point \(C\). A point \(O\) lies on the common tangent at point \(C\), and tangents \(OD\) and \(OE\) are drawn from point \(O\) to the circles centered at \(A\) and \(B\), touching them at points \(D\) and \(E\) respectively. It is given: - \(\angle COD = 56^\circ\) - \(\angle COE = 40^\circ\) - \(\angle ACD = x^\circ\) - \(\angle BCE = y^\circ\) We are to prove: - \(OD = OC = OE\) - \(x - y = 4^\circ\)

10. If two circles touch each other externally at point A, and the number of direct common tangents is \(= x\), and the number of transverse common tangents is \(= y\), then \(x - y =-----\).

11. Two circles touch externally at point \( C \). A common external tangent touches the circles at points \( A \) and \( B \) respectively. Find \( \angle ACB \).

(a) 60° (b) 45° (c) 30° (d) 90°

12. Two fixed circles with centers A and B touch each other internally. Another circle touches the larger circle internally at point X and the smaller circle externally at point Y. If O is the center of this third circle, prove that AO + BO is constant.

13. 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.

14. Two circles touch each other externally. The diameters of the circles are 7 cm and 6 cm respectively. What is the distance between their centers?

(a) 6 cm (b) 13 cm (c) 12.5 cm (d) 6.5 cm

15. Two circles touch each other externally at point A. The common tangent drawn at point A to both circles is a __________ common tangent (direct / transverse).

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 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

18. If two identical circles with a radius of 4 cm are externally tangent to each other, what is the length of their common chord?

19. If two circles centered at P and Q have diameters \(x\) and \(3x\) respectively, and they externally touch each other, find the length of PQ.

(a) \(x\) (b) \(2x\) (c) \(3x\) (d) \(4x\)

20. Three circles with centers \(A\), \(B\), and \(C\) externally touch each other. If \(AB = 5\) cm, \(BC = 7\) cm, and \(CA = 6\) cm, then find the radius of the circle centered at \(C\).

21. The radii of two circles are 5 cm and 3 cm, respectively. If the circles touch each other externally, then the distance between their centers will be –

(a) 2 cm (b) 3 cm (c) 8 cm (d) 1.5 cm

22. 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.

23. Draw an equilateral triangle ABC with each side measuring 5 cm. Then, draw the circumcircle of that triangle. At point A on the circle, draw a tangent. On the tangent, take a point P such that AP = 5 cm. From point P, draw another tangent to the circle, and write down which point on the circle this second tangent touches.

24. On the straight line segment AB, take a point O. From point O, draw a perpendicular PQ to AB. With A and B as centers and radii equal to AO and BO respectively, draw two circles. Write what PQ is called with respect to these two circles. From point P, draw the other tangents to both circles.

25. The radii of two circles are 5 cm and 3 cm respectively. The two circles touch each other externally. Find the distance between the centers of the two circles.

(a) 2 cm (b) 2.5 cm (c) 1.5 cm (d) 8 cm

26. If two circles neither intersect nor touch each other, the maximum number of common tangents that can be drawn to them is _________.

27. In the adjacent figure, three circles with centers A, B, and C touch each other externally. If AB = 5 cm, BC = 7 cm, and CA = 6 cm, then find the radius of the circle centered at A.

28. 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.

29. 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 \(PQ = \frac{1}{2}BC\).

30. 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.