1. A chord is drawn at a distance of 5 cm from the center of a circle with a radius of 10 cm. What is the length of the chord?

(a) \(5\sqrt3\) cm (b) 6 cm (c) 8 cm (d) 3 cm

2. Two identical circles, each with a radius of 13 cm, intersect each other, and the length of their common chord is 10 cm. What is the distance between the centers of the two circles?

3. 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.

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. 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?

6. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.

7. 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.

8. 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

9. 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.

10. AB is a chord (not a diameter) of a circle with center O. If the length of chord AB is 8 cm and the radius of the circle is 5 cm, find the perpendicular distance from the center O to the chord AB.

11. If the perpendicular distance from the center of a circle with a radius of 17 cm to a chord is 4 cm, what is the length of the chord?

12. For a circle centered at \(O\) with a radius of 5 cm and a chord length of 4 cm, the perpendicular distance from \(O\) to the chord is –?

(a) 3 cm (b) 1 cm (c) 4 cm (d) 2 cm

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. In a circle with a radius of 10 cm, if a chord subtends a right angle at the center, then the length of the chord will be 5 cm.

15. 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.

16. Here’s the English translation: Inside a circle with center O, point P is any point. The radius of the circle is 13 cm, and OP = 12 cm. Find the minimum length of a chord that passes through point P.

(a) 5 cm (b) 6 cm (c) 6.5 cm (d) 10 cm

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. Two identical circles, each with a radius of 10 cm, intersect each other, and the length of their common chord is 12 cm. Find the distance between the centers of the two circles.

24. Point P is any point inside a circle centered at O. If the radius of the circle is 5 cm and OP = 3 cm, then find the minimum length of the chord passing through point P.

25. Two parallel chords of a circle with a radius of 5 cm have lengths of 6 cm and 8 cm. What is the distance between the two chords?

26. 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?

27. 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?

28. A hollow cube is made using copper sheets that are 1.5 cm thick, with an external edge length of 20 cm. After melting the cube, how many solid cuboids measuring 7 cm × 3 cm × 1 cm can be made from it?

(a) 200 (b) 220 (c) 100 (d) 147

29. Find the length of the tangent drawn from a point located 13 cm away from the center of a circle with a radius of 5 cm.

30. If a chord of a circle with radius 5 cm has a length of 4 cm, then the distance of the chord from the center is \(\_\_\_\_\) cm.