1. In triangle \(\triangle ABC\), D and E are the midpoints of sides AB and AC respectively. If the area of triangle ABC is 36 square centimeters, then what is the area of triangle ADE?

(a) 6 square centimeters (b) 9 square centimeters (c) 12 square centimeters (d) 16 square centimeters

2. Point O is any point inside parallelogram ABCD. If \(\triangle AOB + \triangle COD = 16\) square cm, then what is the area of parallelogram ABCD?

(a) 8 square cm (b) 4 square cm (c) 32 square cm (d) 64 square cm

3. If E and F are the midpoints of sides AB and AD respectively in parallelogram ABCD, and the area of parallelogram ABCD is 1600 square centimeters, then what is the area of triangle AEF?

(a) 400 sq cm (b) 200 sq cm (c) 300 sq cm (d) None of the above

4. "A, B, C, and D are respectively the midpoints of the sides PQ, QR, RS, and SP of the parallelogram PQRS. If the area of the parallelogram PQRS is 36 square centimeters, then what is the area of the quadrilateral ABCD?"

(a) 24 square cm (b) 18 square cm (c) 30 square cm (d) 36 square cm

5. 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

6. If the quadrilateral formed by joining the midpoints of the sides of parallelogram ABCD has an area of 100 square cm, then what is the area of the parallelogram ABCD?

(a) 400 sq cm (b) 200 sq cm (c) 600 sq cm (d) 800 sq cm

7. In triangle ABC, E and F are the midpoints of sides AB and AC respectively. If the area of triangle AEF is 50 square cm, then what is the area of triangle ABC?

(a) 100 sq cm (b) 200 sq cm (c) 150 sq cm (d) 300 sq cm

8. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC?

(a) 1:2 (b) 2:1 (c) 1:1 (d) None of these

9. 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

10. 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

11. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

12. 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?

13. In triangle \(ABC\) and triangle \(DEF\), \(\angle ABC = \angle DEF\), \(\frac{AB}{DE} = \frac{BC}{EF}\), and the ratio of their areas is \(\frac{\triangle ABC}{\triangle DEF} = \frac{25}{16}\). If \(BC = 10\) cm, what is the length of \(EF\)?

14. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

15. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).

16. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

17. In right-angled triangle ABC, \(\angle B = 90^\circ\), and M is the midpoint of the hypotenuse drawn from point B. If AB = 6 and AC = 8, then what is the length of BM?

(a) 10 (b) 4 (c) 16 (d) None of the above

18. If the base area of a cone is \(x\) square units, its height is \(y\) units, and its volume is \(z\) cubic units, then what is the value of \(\frac{xy}{z}\)?

19. If the total surface area of a hemisphere is \(36\pi\) square centimeters, then its radius will be 3 cm.

20. In right-angled triangle \( \triangle ABC \), where \( \angle A = 90^\circ \), a perpendicular \( AD \) is drawn from point \( A \) to the hypotenuse \( BC \). Prove that: \[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle ACD} = \frac{BC^2}{AC^2} \]

21. If the volume of a right circular cone is \(X\) cubic units, the area of its base is \(Y\) square units, and its height is \(Z\) units, then what is the value of \(\frac{YZ}{X}\)?

22. In triangle PQR, \(\angle\)PQR = 90° and PR = 10 cm. If S is the midpoint of side PR, then the length of QS is?

(a) 4 cm (b) 5 cm (c) 6 cm (d) 3 cm

23. In triangle ABC, point D is the midpoint of side BC, point E is the midpoint of side BD, and point O is the midpoint of line AE. What is the area of triangle BOE?

(a) \(\frac{1}{3} \times\) area of triangle ABC. (b) \(\frac{1}{4} \times\) area of triangle ABC. (c) \(\frac{1}{6} \times\) area of triangle ABC. (d) \(\frac{1}{8} \times\) area of triangle ABC.

24. In triangle ABC, point D lies on side AC such that AD : DC = 3 : 2. If the area of triangle ABC is 40 square cm, then the area of triangle BDC is —

(a) 16 square cm (b) 24 square cm (c) 30 square cm (d) 36 square cm

25. ABC is a right-angled triangle with \(\angle ABC = 90^\circ\); if the coordinates of points A and C are (0, 4) and (3, 0) respectively, then the area of triangle ABC is:

(a) 12 square units (b) 6 square units (c) 24 square units (d) 8 square units

26. If the total surface area of a hemisphere is 36\(\pi\) square centimeters, then the radius will be 3 cm.

27. If the similar triangles \(\triangle\)ABC and \(\triangle\)PQR have AB:PQ = 3:5, then the area ratio of \(\triangle\)ABC to \(\triangle\)PQR is—

(a) 9.25 (b) 25.9 (c) 3.5 (d) 5.3

28. In \( \triangle ABC \), points \( D \) and \( E \) lie on sides \( AB \) and \( AC \) respectively, such that \( DE \parallel BC \) and \( AD:DB = 3:1 \). If \( EA = 33 \) cm, then the length of \( AC \) is _____.

29. If the volume of a right circular cone is \(V\) cubic units, the area of the base is \(A\) square units, and the height is \(H\), then what is the value of \(\frac{AH}{V}\)?

30. We have drawn a median \(AD\) of triangle \( \triangle ABC \). If a straight line parallel to \(BC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that the line segment \(PQ\) is bisected by the median \(AD\).