1. 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

2. 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

3. 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.

4. 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.

5. 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.

6. 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.

7. 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

8. 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.

9. 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.

10. 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

11. Draw a circle with a radius of 4 cm. From a point located 6.5 cm away from the center of the circle, draw two tangents to the circle.

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. 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 \(BC\).

14. 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.

15. 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 \(BC\).

16. 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.

17. 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 \(OA \parallel RQ\).

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. 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

20. Two tangents are drawn to a circle from points P and Q, and they intersect at point A. If ∠PAQ = 80°, then what is the value of ∠APQ?

21. In triangle ABC, ∠A = 90°, AB = 12 cm, AC = 5 cm, and BC = 13 cm. A perpendicular AD is drawn from point A to side BC. What is the length of AD?

22. 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.

23. 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?

24. 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.

25. 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 \(P\)?

26. 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?

27. 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?

28. The diameter of a circle centered at O is 26 cm. The distance from point O to the chord PQ is 5 cm. Calculate and write the length of the chord PQ.

29. 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.

30. AB is a diameter of a circle with center O. P is any point on the circle. Tangents are drawn at points A and B, which are intersected by the tangent drawn from point P at points Q and R respectively. If the radius of the circle is \(r\), prove that \(PQ \cdot PR = r^2\).