1. 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?

2. If the angles of a triangle are in the ratio \(1 : 1 : 2\), then what will be the ratio of the sides of the triangle?

(a) \(1:\sqrt{2}:1 \) (b) \(1:1:\sqrt{2}\) (c) \(1:1:2\) (d) \(1:2:1\)

3. If the ratio of the areas of two similar triangles is 64:49, then find the ratio of their corresponding sides.

4. If the sides of a triangle are in the ratio \(3x : 4x : 5x\), what is the measure of the largest angle of the triangle?

(a) 105\(^o\) (b) 75\(^o\) (c) 60\(^o\) (d) None of the above

5. If the lengths of the three sides of a triangle are in the ratio 3:4:5, then the triangle is a right-angled triangle.

6. 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

7. If the three angles of a triangle are in the ratio 2:3:4, then the measure of the largest angle in degrees is ________.

8. If the three sides of a triangle are in the ratio 5 : 12 : 13, then the triangle will always be a right-angled triangle.

9. Give the definition of a radian. The angles of a triangle are in the ratio 2:5:3; what is the radian measure of the smallest angle of the triangle?

10. \(\triangle\)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.

11. If the ratio of the lengths of two corresponding sides of two similar triangles is 7:11, then the ratio of their perimeters is —

(a) 11:7 (b) 49:121 (c) 7:11 (d) 121:49

12. In \(\triangle\)ABC, if DE \(\parallel\) BC and AD:DB = 3:2, then what is the ratio of DE:BC?

13. If the angles of a triangle are in the ratio 1:1:2, then the sides of the triangle will be in the ratio ———.

14. If the ratio of the lengths of two corresponding sides of two similar triangles is 7:11, then their perimeter ratio is _____.

15. A line parallel to side BC of triangle ABC intersects sides AB and AC at points D and E, respectively. If AD is half of AE, then the ratio BD:EC will be –

(a) 1:2 (b) 2:1 (c) 2:3 (d) 1:3

16. If the three sides of a triangle are in the ratio 3:4:5, then the triangle will always be a right-angled triangle.

17. The volumes of two perpendicular circular rings are equal. The ratio of their heights is 4:9, then what will be the ratio of their radii?

(a) 3:2 (b) 2:3 (c) 4:9 (d) 8:9

18. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points D and E respectively. If \( AB = 20 \) cm and \( BD = 14 \) cm, then what is the ratio \( DE : BC \)?

(a) 7:10 (b) 5:17 (c) 3:10 (d) 7:17

19. 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

20. In a partnership business, Ratan, Suresh, and Ramesh invested in the ratio \(\frac{1}{6} : \frac{1}{5} : \frac{1}{4}\). If the profit at the end of the year is ₹3700, what is Ramesh's share of the profit?

(a) ₹ 1400 (b) ₹ 1200 (c) ₹ 1500 (d) ₹ 1600

21. "If the two acute angles of a right-angled triangle are in the ratio 2:3, what are the radian measures of those two angles?

(a) \(\cfrac{π}{5},\cfrac{3π}{10}\) (b) \(\cfrac{π}{10},\cfrac{3π}{5}\) (c) \(\cfrac{π}{5},\cfrac{3π}{20}\) (d) \(\cfrac{π}{5},\cfrac{π}{15}\)

22. In triangle ABC, \(\angle C = 90^\circ\) and AC : BC = 3 : 4, then what is the value of cosec A?

(a) \(\cfrac{3}{4}\) (b) \(\cfrac{5}{3}\) (c) \(\cfrac{5}{4}\) (d) \(\cfrac{3}{5}\)

23. 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

24. 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

25. If the volumes of two right circular cylinders of equal height are in the ratio 16:25, then what is the ratio of their radii?

(a) 4:5 (b) 5:4 (c) 3:5 (d) 5:3

26. In triangle ABC, if AB\(^2\) + BC\(^2\) = AC\(^2\) and AC = \(\sqrt{2}\) × BC, then what type of triangle is it?

(a) Right-angled (b) Right-angled isosceles (c) Equilateral (d) Isosceles

27. 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

28. 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

29. If the orthocenter, centroid, circumcenter, and incenter of a triangle all lie at the same point, then what type of triangle is it?

(a) isosceles (b) scalene (c) equilateral (d) right-angled isosceles

30. 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