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. 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

3. 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

4. Assume \(PQ\) is the tower and \(AB\) is the stick. \(∴ BC = 4\) cm, \(QC = 28\) meters Triangles \(∆PQC\) and \(∆ABC\) are similar right-angled triangles. Therefore, they are similar. \(∴ \cfrac{PQ}{AB} = \cfrac{QC}{BC} \) Or, \(\cfrac{PQ}{6} = \cfrac{28}{4}\) Or, \(PQ = \cfrac{28 \times 6}{4} = 42\) meters ∴ The height of the tower is 42 meters.

5. 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

6. 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

7. 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?

8. 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?

9. In a rectangular prism, the ratio of length, width, and height is \(4:5:3\). If the length of one diagonal is \(35\sqrt2\) cm, what is the total surface area?

10. 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.

11. The three sides of a triangle are 6 cm, 8 cm, and 10 cm respectively. What is the circumradius of the triangle? This triangle is special—it’s a right triangle (since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). And for right triangles, the circumradius is half the hypotenuse. So the circumradius = \(\frac{10}{2} = 5\) cm.

12. 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?

13. 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

14. 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

15. 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

16. The sum of the length, width, and height of a rectangular box is 24 cm, and the length of its diagonal is 15 cm. What is the total surface area of the rectangular box?

(a) 360 square cm (b) 221 square cm (c) 351quare cm (d) 256 quare cm

17. The base length and width of a rectangular reservoir are 15 meters and 12 meters respectively. Water is filled into the reservoir from a nearby pond using a pump. If the pump can fill 36,000 liters of water per hour, then how long will it take for the pump to fill the reservoir up to a height of 7.2 decimeters? [Note: 1 liter = 1 cubic decimeter]

18. If a right circular cylinder has a radius of 3 cm and a height of 4 cm, then what is the length of the longest rod that can be placed inside it?

19. 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.

20. 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?

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

22. The length, width, and height of a cuboidal room are respectively \(a\), \(b\), and \(c\) units, and given that \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), what is the length of the longest rod that can be placed inside the room?

23. The heights of two pillars are \(h_1\) meters and \(h_2\) meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, and the angle of elevation from the base of the first pillar to the top of the second pillar is 45°, show that \(h_1^2 = 3h_2^2\).

24. The sum of the length, breadth, and height of a rectangular box is 24 cm, and the length of its diagonal is 15 cm. What is the total surface area of the box?

25. 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?

26. 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 _____.

27. 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.

28. In the trapezium ABCD, AB || DC, and P and Q are two points on the sides AD and BC such that PQ || DC. If PD = 18 cm, BQ = 35 cm, and QC = 15 cm, find the length of AD.

29. 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.

30. In \(\triangle\)ABC, \(\angle\)ABC = 90°, and BD \(\bot\) AC. If BD = 6 cm and AD = 4 cm, then what is the length of CD?