1. If a circle has a radius of 10 cm and two parallel chords of lengths 16 cm and 12 cm, what is the perpendicular distance between the two chords?

(a) \(14\sqrt3\) cm (b) 8 cm (c) 2 cm (d) 10 cm

2. AB and CD are two parallel chords, each of length 16 cm. If the radius of the circle is 10 cm, find the distance between the two chords.

(a) 12 cm (b) 16 cm (c) 20 cm (d) 5 cm

3. A circle with a radius of 10 cm has two parallel chords of lengths 4 cm and 6 cm. What is the distance between the two chords?

(a) \(\sqrt7(\sqrt3-\sqrt{12})\) cm (b) \(\sqrt3(\sqrt7+\sqrt{12})\) cm (c) \(\sqrt{13}(\sqrt{12}-\sqrt{7})\) cm (d) \(\sqrt{7}(\sqrt{13}+\sqrt{12})\) cm

4. Two parallel chords AB and CD of lengths 10 cm and 24 cm respectively lie on opposite sides of the center O of a circle. If the distance between the chords AB and CD is 17 cm, find the radius of the circle.

5. 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?

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. 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.

8. 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?

9. Two equal circles, each with a radius of 10 cm, intersect, and the length of their common chord is 12 cm. Determine the distance between the centers of the circles.

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

11. If the lengths of two equal chords are _____ cm and the distance between them is 4 cm, then the diameter of the circle will be 10 cm.

12. In a circle, two chords PQ and PR are perpendicular to each other. If the radius of the circle is \(r\) cm, find the length of the chord QR.

13. In a circle with center O, AB and CD are two equal chords. If \(\angle\)AOB = 60° and CD = 10 cm, then what is the length of OD?

14. The distance between the centers of two circles is 4 cm, and the circles are externally tangent to each other. If the radius of the first circle is 5 cm, determine the radius of the second circle.

(a) 13 cm (b) 6.5 cm (c) 3 cm (d) none of the above

15. Two equal circles, each with a radius of 10 cm, intersect each other. The length of their common chord is 12 cm. Find the distance between their centers.

16. In a circle, two chords PQ and PR are perpendicular to each other. If the radius of the circle is \( r \) cm, find the length of chord QR.

17. Two circles are touching each other externally, and the distance between their centers is 7 cm. If the radius of one circle is 4 cm, find the radius of the other circle.

18. AB and CD are two parallel straight lines. AD and BC intersect each other at point O. If OA = 2 cm, OB = 3 cm, and OD = 4 cm, then what is the length of OC?

(a) 6 cm (b) 4 cm (c) 4.8 cm (d) 4.2 cm

19. 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

20. Two circles with centers A and B have a common tangent PO of length 16 cm, and the length of PB is 10 cm; the area of triangle \(\triangle\)POB will be—

(a) 24 square cm (b) 48 square cm (c) 16 square cm (d) 12 square cm

21. Two circles intersect each other. The radius of each circle is 10 cm. The length of the common chord is 16 cm. Find the distance between the centers of the two circles.

22. In a circle centered at \(O\), two chords \(AB\) and \(CD\) have equal lengths. Given that \(\angle AOB = 60°\) and the radius of the circle is \(6\) cm, find the area of \(\triangle COD\).

(a) \(9\sqrt3\) square c.m (b) \(6\sqrt3\) square c.m (c) \(2\sqrt3\) square c.m (d) \(3\sqrt3\) square c.m

23. O is the center of a circle where AB and CD are two chords of equal length. If the perpendicular distance from O to the chord AB is 4 cm, then the perpendicular distance from O to the chord CD will also be 4 cm.

(a) 2cm (b) 6cm (c) 8cm (d) 4cm

24. In a circle with a radius of 5 cm, AB and AC are two equal chords. The center of the circle is located outside the triangle ABC. If AB = AC = 6 cm, determine the length of the chord BC.

25. The radii of two circles are 8 cm and 3 cm, respectively, and the distance between their centers is 13 cm. Find the length of a common external tangent of the circles.

(a) 10 cm (b) 14 cm (c) 15 cm (d) 12 cm

26. The radii of two circles are 5 cm and 3 cm, respectively. If the circles touch each other externally, then the distance between their centers will be –

(a) 2 cm (b) 3 cm (c) 8 cm (d) 1.5 cm

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. In a circle centered at O, there are two chords of lengths 6 cm and 8 cm. If the distance from the center to the shorter chord is 4 cm, calculate and write the distance from the center to the other chord.

29. The lengths of two sides are 7.6 cm and 6 cm, and the included angle between them is 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

30. If the radii of two circles are 3 cm and 11 cm, and the distance between their centers is 17 cm, find the length of a direct common tangent to the circles.