1. 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.

2. 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.

3. 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

4. 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 \(P\)?

5. 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.

6. 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

7. 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.

8. 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

9. 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.

10. 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?

11. 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.

12. The radius of a circle is 13 cm. If the length of a chord AB in that circle is 10 cm, find the perpendicular distance from the center of the circle to the chord.

13. 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.

14. 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.

15. 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

16. 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

17. The radius of a circle is 13 cm and the length of a chord is 10 cm. What is the distance from the center of the circle to the chord?

(a) 12.5 cm (b) 12 cm (c) \(\sqrt{69}\) cm (d) \(\sqrt{24}\) cm

18. 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?

19. The diameter of a circle centered at O is 26 cm. The perpendicular distance from point O to the chord PQ is 5 cm. Determine the length of the chord PQ.

20. The radius of a circle is 13 cm, and the length of a chord in the circle is 10 cm. Determine the perpendicular distance from the center to the chord.

(a) 12 cm (b) \(\sqrt{69}\) cm (c) 24 cm (d) 12.5 cm

21. 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?

22. 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.

23. 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.

24. The radius of a circle is 13 cm and the length of a chord is 10 cm. Find the distance from the center of the circle to the chord.

(a) 12.5 cm (b) 12 cm (c) √69 cm (d) 24 cm

25. 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.

26. 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

27. 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

28. 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?

29. 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 \(BC\).

30. 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.