1. Diameter PQ of a circle with radius 4 cm and angle ∠PRQ is an angle in the semicircle. Given QR = \(2\sqrt{7}\) cm, find the length of PR = _____

2. In an isosceles triangle ABC, AB = AC. A circle is drawn with AB as the diameter, and it intersects side BC at point D. If BD = 4 cm, find the length of CD.

3. In the isosceles triangle ABC, AB = AC. A circle is drawn using AB as the diameter, and it intersects side BC at point D. If BD = 4 cm, find the length of CD.

4. 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.

5. In the circle with center \( O \), if \( AOB \) is a diameter and point \( C \) is on the circle such that \( AC = 3 \) cm and \( BC = 4 \) cm, what is the length of \( AB \)?

(a) 3 cm (b) 4 cm (c) 5 cm (d) 8 cm

6. 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

7. 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.

8. 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?

9. 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.

10. 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

11. 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.

12. In isosceles triangle ABC, the circumcenter is O and \(\angle ABC = 120^\circ\). If the radius of the circle is 5 cm, find the length of side AB.

13. In the adjacent figure, BX and CY are the bisectors of \(\angle\)ABC and \(\angle\)ACB respectively. Given that AB = AC and BY = 4 cm, find the length of AX.

14. 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

15. AOB is the diameter of a circle. If AC = 3 cm and BC = 4 cm, find the length of AB.

(a) 3 cm (b) 4 cm (c) 5 cm (d) 8 cm

16. 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.

17. 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.

18. In the adjacent figure, triangle ABC is inscribed in a circle and touches the circle at points P, Q, and R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, and BC = x cm, then find the value of x.

19. Here’s the English translation: Inside a circle with center O, point P is any point. The radius of the circle is 13 cm, and OP = 12 cm. Find the minimum length of a chord that passes through point P.

(a) 5 cm (b) 6 cm (c) 6.5 cm (d) 10 cm

20. In a circle with center \(O\), \(AB\) is the diameter, and \(P\) is a point on the circle. If \(\angle AOP = 104°\), find the value of \(\angle BPO\).

(a) 54° (b) 72° (c) 36° (d) 27°

21. If a triangle similar to one with sides 4 cm, 6 cm, and 8 cm has its largest side measuring 6 cm, what is the length of the smallest side of that triangle?

(a) 4 cm (b) 3 cm (c) 2 cm (d) 5 cm

22. In triangle ABC, \(\angle\)BAC = 90°, and AD is perpendicular to BC. Given: AC = 8 cm, AB = 6 cm Find: The length of BD.

(a) 6 cm (b) 1.5 cm (c) 3 cm (d) 3.6 cm

23. In triangle ABC, \(\angle\)BAC = 90° and AD is perpendicular to BC. Given: AD = 8 cm, BC = 20 cm, and CD > BD Find: The length of CD.

(a) 6 cm (b) 4 cm (c) 20 cm (d) 16 cm

24. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

25. ABCD is a rectangle. O is the point where the diagonals intersect. If AB = 4 cm and OD = 2.5 cm, then what is the length of BC?

(a) 4 cm (b) 1.5 cm (c) 3 cm (d) 2 cm

26. 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

27. In the adjacent figure, BX and CY are the bisectors of ∠ABC and ∠ACB respectively. Given that AB = AC and BY = 4 cm, find the length of AX.

(a) 4 cm (b) 8 cm (c) 6 cm (d) 10 cm

28. 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.

29. 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.

30. 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.