1. 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

2. 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

3. 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

4. From point A of triangle ABC, a perpendicular AD is drawn to side BC, meeting at point D. If BD = 8 cm, DC = 2 cm, and AD = 4 cm, then write the measure of ∠BAC.

5. 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.

6. 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

7. 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

8. In triangle ABC, ∠A = 90°, AB = 12 cm, AC = 5 cm, and BC = 13 cm. A perpendicular AD is drawn from point A to side BC. What is the length of AD?

9. 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.

10. 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?

11. In \(\triangle ABC\), the center of the incircle is \(O\), and the incircle touches the sides \(AB\), \(BC\), and \(CA\) at points \(P\), \(Q\), and \(R\) respectively. Given that \(AP = 4\) cm, \(BP = 6\) cm, \(AC = 12\) cm, and \(BC = x\) cm, determine the value of \(x\).

12. 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.

13. 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.

14. 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

15. In right-angled triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \) and BD is perpendicular to AC. If BD = 8 cm and AD = 4 cm, what is the length of CD?

16. In a circle with center O, a tangent PT is drawn from an external point P to the circle, with T being the point of tangency. If PT = 12 cm and OP = 13 cm, the diameter of the circle will be:

(a) 5 cm (b) 8 cm (c) 6 cm (d) 10 cm

17. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP : PB = 2 : 1 \) and \( AC = 18 \) cm, find the length of \( AQ \).

(a) 12 cm (b) 9 cm (c) 6 cm (d) None of the above

18. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP = 18 \) cm, \( QC = 9 \) cm, and \( AQ = 2 \times PB \), find the length of \( PB \).

(a) 6 cm (b) 12 cm (c) 18 cm (d) 9 cm

19. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

20. 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

21. "Line segments AB and PQ intersect at point O. AP and BQ are perpendiculars to AB. OA = 20 cm, OB = 8 cm, AP = 10 cm. Find the length of BQ."

(a) 4 cm (b) 6 cm (c) 8 cm (d) None of the above

22. 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

23. 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

24. 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

25. 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

26. In triangle ∆ABC, a straight line parallel to side BC intersects AB and AC at points P and Q respectively. Given that \(\frac{AQ}{QC} = \frac{3}{4}\) and AB = 21 cm, find the length of PB.

27. In triangle ∆ABC, a straight line parallel to side BC intersects sides AB and AC at points X and Y respectively. If AX = 2.4 cm, AY = 3.2 cm, and YC = 4.8 cm, find the length of AB.

28. 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?

29. 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.

30. A wooden box with a lid is made using wood that is 0.5 cm thick. The external dimensions of the box are: length = 20 cm, width = 16 cm, and height = 12 cm. What is the volume of the wood used to make the box?

(a) 800 cubic centimeters (b) 790 cubic centimeters (c) 820 cubic centimeters (d) 850 cubic centimeters