1. In a circle, two chords AB and AC are perpendicular to each other. If AB = 4 cm and AC = 3 cm, find the radius of the circle.

2. 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.

3. In a circle centered at \( O \), the chords \( AB \) and \( AC \) are perpendicular to each other. If \( AB = 8 \) cm and \( AC = 6 \) cm, determine the radius of the circle.

(a) 7 cm (b) 10 cm (c) 5 cm (d) 9 cm

4. In a circle, AB and AC are two mutually perpendicular chords. If AB = 4 cm and AC = 3 cm, then the radius of the circle is _____ cm.

5. 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.

6. In a circle centered at O, the chords AB and AC are perpendicular to each other. If AB = 6 cm and AC = 8 cm, then the radius of the circle will be -.

(a) 7 cm (b) 10 cm (c) 5 cm (d) 4 cm

7. In a circle with a radius of 5 cm, the chords AB and BC are perpendicular to each other. Find the length of the chord AC.

(a) 5 cm (b) 2.5 cm (c) 10 cm (d) none of the above

8. 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

9. QR is a chord of a circle, and POR is a diameter of the circle. OD is perpendicular to chord QR. If OD = 4 cm, find the length of PQ.

(a) 4cm (b) 2cm (c) 8cm (d) none of the above

10. 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.

11. In the adjacent figure, O is the center of the circle, and OP is perpendicular to AB. If AB = 6 cm and PC = 2 cm, then find the radius of the circle.

12. QR is a chord of the circle, and POR is a diameter of the circle. OD is perpendicular to the chord QR. If OD = 4 cm, then the length of PQ is –

(a) 4 cm (b) 2 cm (c) 8 cm (d) None of these

13. 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

14. AB is the diameter of a circle, and PQ is a chord that stands perpendicular to AB at point O. Given: OA = 8 cm, OB = 2 cm, OP = 4 cm Find: The length of OQ.

(a) 6 cm (b) 4 cm (c) 5 cm (d) None of the above

15. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

16. In right-angled triangle ABC, ∠ABC = 90°, AB = 3 cm, BC = 4 cm, and from point B, a perpendicular BD is drawn to side AC, meeting AC at point D. Find the length of BD.

17. 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

18. Three circles with centers \(A\), \(B\), and \(C\) externally touch each other. If \(AB = 5\) cm, \(BC = 7\) cm, and \(CA = 6\) cm, then find the radius of the circle centered at \(C\).

19. In a circle with a radius of 5 cm, AB and AC are two equal-length chords. The center of the circle lies outside triangle ABC. Given AB = AC = 6 cm, find the length of chord BC.

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

21. The radius of the first sphere \( = \cfrac{21}{2} \) cm And the radius of the second sphere \( = \cfrac{17.5}{2} \) cm \( = \cfrac{175}{2 \times 10} \) cm \( = \cfrac{35}{4} \) cm

The ratio of the amount of metal used to make the two spheres = Ratio of their surface areas \( = 4π\left(\cfrac{21}{2}\right)^2 : 4π\left(\cfrac{35}{4}\right)^2 = \cfrac{21 \times 21}{4} : \cfrac{35 \times 35}{16} \) \( = 9 : \cfrac{25}{4} = 36 : 25 \) (Answer)

22. Two chords AB and CD of a circle with center O intersect each other at point P. Prove that \(\angle AOD + \angle BOC = 2\angle BPC\) If \(\angle AOD\) and \(\angle BOC\) are supplementary, then prove that the two chords are perpendicular to each other.

23. In the right-angled triangle ABC, ∠ABC = 90°, AB = 3 cm, BC = 4 cm, and from point B a perpendicular BD is drawn to side AC, meeting it at point D. Find the length of BD.

24. In the adjacent figure, a circle is centered at point O. From an external point C, two tangents are drawn to the circle, touching it at points P and Q respectively. Another tangent is drawn at point R on the circle, which intersects the tangents CP and CQ at points A and B respectively. If CP = 11 cm and BC = 7 cm, then find the length of BR.

25. A circle has a radius of 2.5 cm. Two chords AB = 4 cm and AC = 3 cm are drawn. What is the measure of angle ∠BAC?