1. 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.

2. 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.

3. 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 \(P\)?

4. 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

5. 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.

6. 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?

7. 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?

8. 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?

9. I have drawn a circle with center O and radius 6 cm. From a point P located 10 cm away from the center O, a tangent PT is drawn to the circle. Calculate and write the length of the tangent PT.

10. 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

11. 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?

12. 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?

13. Let's draw a circle with a radius of 2.5 cm. Take a point outside the circle that is 6.5 cm away from the center. Then, draw a tangent to the circle from that external point and measure the length of the tangent using a ruler.

14. 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.

15. 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

16. 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?

17. 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 16 cm?

18. 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

19. The radius of a circle centered at point O is 5 cm. Point P lies at a distance of 13 cm from point O. From point P, two tangents PQ and PR are drawn to the circle. Find the area of quadrilateral PQOR.

(a) 60 square cm (b) 30 square cm (c) 120 square cm (d) 150 square cm

20. The radius of a circle with center O is 5 cm. From an external point P, which is located at a certain distance from point O, two tangents PQ and PR are drawn to the circle. The quadrilateral PQOR has an area of 60 square centimeters. Find the distance from point O to point P.

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. A tangent is drawn from an external point A to a circle centered at O, touching the circle at point B. Given: OB=5 cm, AO=13 cm, Find the length of AB.

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

23. The radius of a circle with center \(O\) is 5 cm. Point \(P\) is located at a distance of 13 cm from \(O\). From point \(P\), two tangents \(PQ\) and \(PR\) are drawn to the circle. Find the area of the quadrilateral \(PQOR\).

(a) \(60\) square cm (b) \(30\) square cm (c) \(120\) square cm (d) \(150\) square cm

24. A circle has center O and a radius of 5 cm. AB is a chord of the circle with a length of 8 cm. Calculate and write the distance from point O to the chord AB.

25. 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

26. A circle is centered at O with a radius of 6 cm. From a point P located 10 cm away from the center O, determine the length of the tangent PT, where T is the point of tangency.

27. 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.

28. Niyamat has drawn a circle with a radius of 13 cm. I have drawn a chord AB of length 10 cm in this circle. Calculate and write the distance from the center of the circle to this chord AB.

29. Let’s draw a circle with a radius of 2.8 cm. Then, take a point that is 7.5 cm away from the center of the circle. From that external point, draw two tangents to the circle.

30. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?