1. A chord of length 4 2 meters subtends a right angle at the center of a circle. What is the radius of the circle?

(a) \(4\sqrt2\) meters (b) \(2\sqrt2\) meters (c) \(2\) meters (d) \(4\) meters

2. What is the length of a chord of a circle with a radius of 10 cm, if the chord subtends a right angle at the center?

3. 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.

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. The length of a chord of a circle is 14 cm, and the perpendicular distance from the center to the chord is 24 cm. Find the radius of the circle.

6. 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.

7. 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.

8. The radius of the cross-section of a solid right circular rod is 3.2 decimeters. The rod was melted and recast into 21 solid spheres, each with a radius of 4 centimeters. If so, calculate and write the length of the rod.

9. A chord 18 cm long in a circle subtends a right angle (90°) at the center of the circle. What is the perpendicular distance from the center to the chord?

(a) \(6\sqrt{2}\) cm (b) 6 cm (c) 9 cm (d) \(9\sqrt{2}\) cm

10. A horse runs along a circular path with a radius of 21 meters. In 5 seconds, it covers an arc that subtends a 60° angle at the center of the circle. Find the speed of the horse.

11. AB is a diameter of a circle with center O. CD is a chord whose length is equal to the radius of the circle. When AC and BD are extended, they intersect at point P. What is the measure of ∠APB?

12. 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.

13. If the curved surface area of a right circular cylinder is 264 square meters and its volume is 924 cubic meters, then what is the radius of the base?

14. The radius of a circle is 28 cm. What is the radian measure of the central angle subtended by an arc of length 5.5 cm on this circle?

15. In a semicircle with a radius of 4 cm, AB is the diameter and ∠ACB is an angle inscribed in the semicircle. If BC = \(2\sqrt{7}\) cm, find the length of AC.

16. A one-end open cylindrical pipe is 12 meters long and has a radius of 4.5 meters. What is the length of the longest rod that can be placed inside the cylinder?

17. Perfectly explained. The triangle formed is a right-angled isosceles triangle, where the angle of elevation is 45°, making the opposite and adjacent sides equal. So, the height of the pillar equals the distance from its base: 20 meters.

(a) 1:2:3 (b) 1:5:7 (c) 1:7:9 (d) None of the above

18. Given: In a circle with center O, MN is any chord that is not a diameter, and AC is a diameter. To Prove: AC > MN, i.e., the diameter is the longest chord of a circle. Construction: From the center O, draw a perpendicular OD to the chord MN. Join O and M. Proof: OM > MD [∵ Triangle OMD is a right-angled triangle and OM is the hypotenuse] Or, OA > MD [∵ OA = OM, both are radii of the same circle] Or, ½AC > ½MN Or, AC > MN Therefore, the diameter is the longest chord of a circle.

19. If the radius of a circle is \(r\sqrt{2}\) and the perpendicular distance from the center to a chord is \(r\), then what is the length of the chord?

20. If two identical circles with a radius of 4 cm intersect such that their centers lie on a straight line, what is the length of their common chord?

(a) \(4\sqrt3\) cm (b) \(5\sqrt3\) cm (c) \(6\sqrt3\) cm (d) \(7\sqrt3\)

21. 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

22. 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.

23. In a circle centered at O, the chords AB and CD are of equal length. If \(\angle\)AOB = 60° and CD = 6 cm, determine the radius of the circle.

24. 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?

25. A cylindrical water tank with a length of 10.5 meters is connected to a pipe with a diameter of 7 cm. If water flows through the pipe at a speed of 210 meters per minute and the tank empties in 45 minutes, what is the radius of the tank?

26. 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

27. The radius of a circle is 28 cm. What is the circular measure of the central angle subtended by an arc of length 5.5 cm in that circle?

28. “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.”

29. A flag of 3.3 meters in length is mounted on the roof of a three-story building. From a certain point on the street, the angles of elevation to the top and bottom of the flagpole are 50° and 45°, respectively. Determine the height of the building. [tan 50° = 1.192]

30. The English translation is: "When the sun's angle of elevation increases from 45° to 60°, the length of the shadow of a pole decreases by 3 meters. Determine the height of the pole. (Take \(\sqrt{3} = 1.732\))"