1. 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.

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. 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.

4. 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.

5. If two circles with radii of 10 cm and 7 cm are internally tangent to each other, the distance between their centers will be—

(a) 4 cm (b) 6 cm (c) 17 cm (d) 3 cm

6. 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

7. 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.

8. In triangle \(ABC\) and triangle \(DEF\), \(\angle ABC = \angle DEF\), \(\frac{AB}{DE} = \frac{BC}{EF}\), and the ratio of their areas is \(\frac{\triangle ABC}{\triangle DEF} = \frac{25}{16}\). If \(BC = 10\) cm, what is the length of \(EF\)?

9. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent between the two circles.

10. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent to the two circles.

11. 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.

12. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of a side of the first triangle is 4 cm, then the length of the corresponding side of the second triangle will be ____ .

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. The outer dimensions of a box with a lid are 12 cm in length, 10 cm in width, and 8 cm in height. The total inner surface area of the box is 376 square cm. If the walls of the box have equal thickness, what is their thickness?

15. “The perimeters of two similar triangles are 24 cm and 16 cm respectively. If one side of the second triangle is 6 cm, what will be the length of the corresponding side of the first triangle.”

16. 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.

17. 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.

18. ABC is a right-angled triangle with \(\angle B\) being the right angle and BD ⟂ AC. If AD = 4 cm and CD = 16 cm, then calculate and write the lengths of BD and AB.

19. The radii of two circles are 8 cm and 3 cm respectively, and the distance between their centers is 1.3 cm. Find the length of a direct common tangent between the two circles.