1. From an external point A, a tangent AB is drawn to a circle centered at O, touching the circle at point B. If OB = 5 cm and AO = 13 cm, find the length of AB.

(a) 12 cm (b) 13 cm (c) 6.5 cm (d) 6 cm

2. 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.

3. From a point outside a circle, a tangent of length 8 cm is drawn to the circle. If the radius of the circle is 6 cm, what is the distance from the center of the circle to the external point?

(a) 10 cm (b) 8 cm (c) 6 cm (d) 14 cm

4. From an external point located 17 cm away from the center of a circle with a diameter of 16 cm, what is the length of the tangent drawn to the circle?

5. If a circle has a radius of 5 cm and a tangent is drawn from an external point \(P\) to the circle with a length of 12 cm, what is the distance from the center to point \(P\)?

6. From an external point \(P\), a tangent is drawn to a circle centered at \(O\), touching the circle at \(R\). Given \(OR = 15\) cm and \(PR = 8\) cm, find \(OP\).

(a) 17 cm (b) 7 cm (c) 23 cm (d) 20 cm

7. 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

8. From an external point \(A\), two tangents are drawn to a circle centered at \(O\), touching the circle at points \(B\) and \(C\) respectively. Prove that the line \(AO\) is the perpendicular bisector of the line segment \(BC\).

9. A circle has center ‘O’, and a point P lies 26 cm away from it. If the length of the tangent drawn from point P to the circle is 10 cm, then what is the radius of the circle?

10. From an external point \(A\), two tangents are drawn to a circle centered at \(O\), touching the circle at points \(B\) and \(C\) respectively. Prove that the line \(AO\) is the perpendicular bisector of the chord \(BC\).

11. From an external point \(A\), two tangents \(AP\) and \(AQ\) are drawn to a circle centered at \(O\), touching the circle at points \(P\) and \(Q\) respectively. If \(PR\) is a diameter of the circle, prove that \(OA \parallel RQ\).

12. Point P is an external point to a circle with center O. The distance from point P to the center of the circle is 26 cm, and the length of the tangent drawn from point P to the circle is 10 cm. The radius of the circle is ____ cm.

13. From an external point P, two tangents PA and PB are drawn to a circle centered at O. If PA = 9 cm and ∠APB = 60°, then what are the measures of ∠PAB and the length of chord AB?

(a) 9 cm (b) 3 cm (c) 6 cm (d) 12 cm

14. From an external point \(P\), two tangents \(PA\) and \(PB\) are drawn to a circle centered at \(O\). Given \(PA = 9\) cm and \(\angle APB = 60°\), find the length of \(AB\).

15. If a circle is centered at point 'O' with a radius of 13 cm and chord AB has a length of 10 cm, what is the distance from point 'O' to chord AB?

16. The radius of a circle centered at point O is 5 cm, and AB is a chord of length 8 cm. Calculate and write the distance from point O to the chord AB.

17. The length of chord PQ in a circle centered at O is 4 cm, and the distance from point O to PQ is 2.1 cm. Calculate and write the diameter of the circle.

18. If I draw a circle centered at point O, and from a point P located 26 cm away from the center a tangent is drawn to the circle which measures 10 cm in length, then calculate and write the length of the radius of the circle.

19. 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: - \(OD = OC = OE\) - \(x - y = 4^\circ\)

20. From an external point A of a circle, two tangents AB and AC are drawn, touching the circle at points B and C respectively. A tangent is drawn at point X, which lies on the arc BC, and it intersects AB and AC at points D and E respectively. Prove that the perimeter of triangle ∆ADE = 2 × AB.

21. In a circle with center O, a tangent PT is drawn from an external point P to the circle, with T being the point of tangency. If PT = 12 cm and OP = 13 cm, the diameter of the circle will be:

(a) 5 cm (b) 8 cm (c) 6 cm (d) 10 cm

22. The distance from the center of a circle to an external point is 13 cm. The length of the tangent from that point to the circle is 12 cm. The radius of the circle is—

(a) 25 cm (b) 1 cm (c) 5 cm (d) \(\cfrac{13}{12}\) cm

23. From point A, which is located 26 cm away from the center O of a circle, a tangent is drawn to the circle with a length of 10 cm. Find the radius of the circle.

24. What is the length of the tangent drawn from an external point located 17 cm away from the center of a circle with a radius of 4 cm?

25. Prove that the two tangents drawn from an external point to a circle are equal in length from the point to the points of contact on the circle.

26. The radius of a circle centered at point O is 5 cm. Point P is located 13 cm away from point O. PQ and PR are two tangents drawn from point P to the circle. What is the area of quadrilateral PQOR?

27. Prove that the two tangents drawn to a circle from an external point are equal in length, and the line segments joining the points of contact to the external point subtend equal angles at the center of the circle.

28. A circle is centered at point O with a radius of 10 cm. A perpendicular is drawn from O to a chord AB, and the length of this perpendicular is 6 cm. What is the length of the chord AB?

29. AB is a chord of a circle with center O. From point O, a perpendicular OP is drawn to the chord AB. The extended line OP intersects the circle at point C. If AB = 6 cm and PC = 1 cm, then what is the radius of the circle?

30. The maximum number of tangents that can be drawn to a circle from an external point is _____.