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. 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

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. 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

6. 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.

7. 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 \(BC - XY\)?

8. 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

9. "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

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. 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

12. "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

13. 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.

14. 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.

15. 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 \(EF\)?

16. What is the radius of the circumcircle of a triangle whose sides are 20 cm, 21 cm, and 29 cm?

(a) 14\(\frac{1}{2}\) cm (b) 14 cm (c) 10 cm (d) 11\(\frac{1}{2}\) cm

17. The three sides of a triangle are 4 cm, 5 cm, and 7 cm respectively. Therefore, the triangle is –

(a) acute-angled (b) obtuse-angled (c) right-angled (d) None of the above

18. 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

19. 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

20. 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

21. 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

22. A rectangular playground is 45 meters long and 40 meters wide. Around it, there is a road of uniform width along all four sides. The area of the road is 450 square meters. What is the width of the road?

(a) 5 meter (b) 2.5 meter (c) 1.5 meter (d) 4.5 meter

23. Draw a triangle in which two sides are 9 cm and 7 cm, and the included angle between them is 60°. Then draw the incircle of that triangle. (Only construction marks are required.)

24. 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.

25. 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.

26. 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.

27. 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.

28. 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.

29. 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) \(\frac{1}{4}\)

30. 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