1. 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

2. 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

3. 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

4. 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?

5. "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

6. In the parallelogram ABCD, point P is the midpoint of side CD. If the area of triangle \( \triangle APD \) is 25 square centimeters, then what is the area of the parallelogram?

(a) 100 square cm (b) 75 square cm (c) 150 square cm (d) 50 square cm

7. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm respectively. If the triangle is rotated once completely around the longer of the two adjacent sides, find the total surface area and the volume of the solid formed.

8. 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 \(BC - XY\)?

9. In triangle ABC, the midpoints of sides BC, CA, and AB are D, E, and F respectively. BE and DF intersect at point X, and CF and DE intersect at point Y. Then, the length of XY is equal.

(a) \(\cfrac{1}{2}\)BC (b) \(\cfrac{1}{4}\)BC (c) \(\cfrac{1}{3}\)BC (d) \(\cfrac{1}{8}\)BC

10. 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

11. 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

12. Draw a square with the same area as an isosceles triangle whose base length is 7 cm and each of the equal sides is 5 cm.

13. 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.

14. 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

15. 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

16. Two circles with centers A and B have a common tangent PO of length 16 cm, and the length of PB is 10 cm; the area of triangle \(\triangle\)POB will be—

(a) 24 square cm (b) 48 square cm (c) 16 square cm (d) 12 square cm

17. If the length, breadth, and height of a right rectangular prism are in the ratio 3:2:1 and its total surface area is 88 square centimeters, then what is its volume?

(a) 120 cubic cm (b) 64 cubic cm (c) 48 cubic cm (d) 24 cubic cm

18. If the area of the square drawn on one side of any triangle is equal to the sum of the areas of the squares drawn on the other two sides, then prove that the angle opposite to the first side is a right angle.

19. If the bases of two triangles lie on the same straight line and the other vertex of both triangles is common, then the ratio of their areas is ______to the ratio of the lengths of their bases.

20. Prove that in a triangle, if the area of the square constructed on one side is equal to the sum of the areas of the squares constructed on the other two sides, then the angle opposite the longest side is a right angle.

21. In a right-angled triangle, the hypotenuse is 6 cm longer than one of the other two sides and 12 cm longer than the other. Find the area of the triangle.

22. \(\triangle\)ABC ~ \(\triangle\)DEF; BC and EF are corresponding sides. If BE : EF = 1 : 3, then the ratio of the areas of \(\triangle\)ABC and \(\triangle\)DEF will be 1 : 27.

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. If the area, perimeter, and height of an equilateral triangle are \(a\), \(s\), and \(h\) respectively, then the value of \(\dfrac{2a}{sh}\) is —

(a) 1 (b) \(\frac{1}{2}\) (c) \(\frac{1}{3}\) (d) \(\frac{1}{4}\)

25. The lengths of the two equal sides of an isosceles triangle are 5 cm, and the base is 6 cm. The area of the triangle is —

(a) 18 square cm (b) 12 square cm (c) 15 square cm (d) 30 square cm

26. "The differences between the semi-perimeter of a triangle and the lengths of its sides are respectively 4 cm, 7 cm, and 5 cm. The area of the triangle is –"

(a) \(20\sqrt{7}\) square cm (b) \(10\sqrt{14}\) square cm (c) \(20\sqrt{14}\) square cm (d) 140 square cm

27. 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

28. A hollow cylindrical iron pipe open at both ends has an inner radius and outer radius of 4 cm and 5 cm respectively. If the total surface area of the pipe is 1188 square centimeters, what is the length of the pipe?

29. 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

30. 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?