1. 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

2. 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

3. 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

4. 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

5. ABC and POR are two similar triangles. If BC = 5 cm, QR = 4 cm, and the height AD = 3 cm, then what is the length of the height PE?

(a) 4.2 cm (b) 1.25 cm (c) 5.4 cm (d) 2.4 cm

6. 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?

7. In a circle with center O, AB and CD are two equal chords. If \(\angle\)AOB = 60° and CD = 10 cm, then what is the length of OD?

8. In the 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 what is the value of x?

9. In \(\triangle\)ABC, a line parallel to side BC intersects AB and AC at points P and Q, respectively. If AP = 4 cm, QC = 9 cm, and PB = AQ, then what is the length of PB?

10. In \(\triangle\)ABC, \(\angle\)ABC = 90° and BD \(\bot\) AC; if AB = 6 cm, BD = 3 cm, and CD = 5.4 cm, then calculate the length of side BC.

11. In the adjacent figure, point O is located inside triangle PQR such that ∠POR = 90°, OP = 6 cm and OR = 8 cm. If PR = 24 cm and ∠QPR = 90°, then write the length of side QR.

12. 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.

13. In triangle \( \triangle ABC \), if \( \angle ABC = 90^\circ \), \( AB = 6 \) cm, and \( BC = 8 \) cm, then what is the length of the circumradius of triangle \( \triangle ABC \)?

14. 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.

15. In triangle ABC, ∠B = 90° and BD ⊥ AC; If AB = 30 cm, BD = 24 cm, and AD = 18 cm, then what is the length of BC?

16. 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

17. 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

18. 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

19. 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

20. In the rhombus PQRS, if ∠Q = 60° and RS = 6 cm, what is the length of the diagonal PR?

(a) \(6\sqrt3\) cm (b) \(\sqrt3\) cm (c) \(\cfrac{1}{\sqrt3}\) cm (d) 6 cm

21. 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

22. 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?

23. In triangle △ABC, if ∠ABC = 90°, AB = 5 cm, and BC = 12 cm, then what is the radius of its circumcircle?

24. In a circle centered at O, chords AB and CD are equidistant from the center. If ∠AOB = 60° and CD = 6 cm, then what is the radius of the circle?

25. In triangle △ABC, ∠ABC = 90° and BD ⊥ AC. If AB = 5 cm and BC = 12 cm, then what is the length of BD?

26. 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.

27. 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?

28. 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

29. In \(\triangle\)ABC, \(\angle\)ABC = 90° and BD \(\perp\) AC. If BD = 6 cm and AD = 4 cm, then find the length of CD.

30. AOB is the diameter of a circle centered at O. C is a point on the circumference. If AC = 3 cm and BC = 4 cm, then the length of AB will be _____.