1. 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

2. 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

3. 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

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. 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. 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

7. 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.

8. 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

9. In two concentric circles, a chord of the larger circle is tangent to the smaller circle. If the radii of the two circles are 10 cm and 4 cm respectively, then what is the length of the chord?

(a) 8 cm (b) 9 cm (c) 11 cm (d) 12 cm

10. 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

11. 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.

12. AB is a chord of a circle with center O. From point O, a perpendicular OP is drawn to the chord AB. The extended line OP intersects the circle at point C. If AB = 6 cm and PC = 1 cm, then what is the radius of the circle?

13. ABC is a right-angled triangle with hypotenuse BC. From point A, a perpendicular AD is drawn to BC. If BD = 4 cm and DC = 5 cm, then what is the length of AB?

14. AB is a chord of a circle centered at point O. OP is perpendicular to AB, where AB = 8 cm and OP = 3 cm. Find the diameter of the circle.

15. 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.

16. 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.

17. The diameter of a circle is 20 cm. If the perpendicular distance from the center to a chord is 4 cm, determine the length of the chord.

18. 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.

19. The diameter of a circle is 20 cm. If the perpendicular distance from the center of the circle to a chord is 4 cm, find the length of the chord.

20. 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.

21. 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.

22. In the adjacent figure, ∠ABC = 90° and BD is perpendicular to AC; if AB = 30 cm, BD = 24 cm, and AD = 18 cm, then write the length of BC.

23. In the adjacent figure, ∠ABC = 90° and BD is perpendicular to AC; if BD = 8 cm and AD = 4 cm, then write the length of CD.

24. Triangle ABC is inscribed in a circle, and the circle touches the sides at points P, Q, and R respectively. If AP = 4 cm, BP = 6 cm, and AC = 12 cm, then find the length of BC.

25. 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.

26. 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?

27. 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.

28. Point P is any point inside a circle centered at O. If the radius of the circle is 5 cm and OP = 3 cm, then find the minimum length of the chord passing through point P.

29. 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.

30. Prove that if a chord (which is not a diameter) of a circle is bisected by a straight line drawn from the center of the circle, then that line is perpendicular to the chord.