1. 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

2. 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

3. Given two parallel chords AB and CD, each of length 16 cm, and the radius of the circle is 10 cm, what is the distance between the two chords?

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

4. 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.

5. 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

6. 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.

7. 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.

8. 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.

9. 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

10. 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.

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. 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?

13. 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

14. 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.

15. 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.

16. 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.

17. 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.

18. 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).

19. The radii of two circles are 5 cm and 3 cm. If the two circles are externally tangent to each other, the distance between their centers will be -

(a) 2 cm (b) 2.5 cm (c) 1.5 cm (d) 8 cm

20. The radii of two circles are 5.5 cm and 2 cm. The circles touch each other internally. The distance between the centers of the two circles is —

(a) 3.5 cm (b) 2.5 cm (c) 1.5 cm (d) 7.5 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. If two circles with radii of 7 cm and 3 cm touch externally, then the distance between their centers will be 4 cm.

23. The radii of two circles are 3.5 cm and 2 cm. The circles touch each other internally. The distance between the centers of the two circles will be –

(a) 5.5 cm (b) 1 cm (c) 1.5 cm (d) None of the above

24. 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.

25. 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.

26. 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

27. 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

28. Draw a triangle with side lengths of 7 cm, 6 cm, and 5.5 cm. Draw the incircle of the triangle and measure the radius of the incircle.

29. 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.

30. 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.