1. If two triangles have their bases on the same straight line and share the same vertex (the opposite vertex), then the ratio of their areas is equal to the ratio of the lengths of their bases.

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

3. 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

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

5. If the diameters of two cones are equal and the ratio of their slant heights is 5:7, then what is the ratio of their curved surface areas?

(a) 25:7 (b) 25:49 (c) 5:49 (d) 5:7

6. If the ratio of the radii of two tall solid circular cones is 2:3 and the ratio of their heights is 5:3, then what is the ratio of their curved surface areas?

(a) 8:07 (b) 2:05 (c) 10:09 (d) 16:09

7. If two triangles are equiangular (have equal corresponding angles), then the ratios of their corresponding sides will be equal; that is, their corresponding sides will be proportional.

8. If a solid cone and a solid right circular cylinder have the same base radius and same height, and the ratio of their curved surface areas is 5:8, then determine the ratio of their base radius to height.

9. If two solid right circular cones have equal volume and their heights are in the ratio 1:2, then what is the ratio of the lengths of their radii?

(a) \(1:\sqrt2\) (b) \(1:2\) (c) \(\sqrt2 : 1\) (d) \(2:1\)

10. If the ratio of the radii of two vertical solid circular cylinders is 2:3, and the ratio of their heights is 5:3, then the ratio of their curved surface areas will be –

(a) 2:5 (b) 8:7 (c) 10:9 (d) 16:9

11. If the volumes of two cones are in the ratio 1:4 and the ratio of the diameters of their bases is 4:5, find the ratio of their heights

(a) 1:5 (b) 5:4 (c) 5:16 (d) 25:64

12. If a hemisphere and a cone have equal base areas and equal heights, then what will be the ratio of their volumes?

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

13. The ratio of the base radii of two right circular cones is 2 : 3, and the ratio of their heights is 3 : 2. Then the ratio of their volumes is —

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

14. If the volumes of two cones are in the ratio 2:3 and the ratio of their base radii is 1:2, then what is the ratio of their heights?

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

15. A hemisphere and a cone have the same base and height. Determine the ratio of their volumes and the ratio of their curved surface areas.

16. If a hemisphere and a cone have the same base area and the same height, find the ratio of their curved surface areas.

17. A solid sphere and a solid cone have the same diameter, and their total surface areas are equal. What is the ratio of the cone's height to its diameter?

18. Two friends start a partnership business by investing ₹40,000 and ₹50,000 respectively. They agree that 50% of the profit will be shared equally, and the remaining 50% will be divided in the ratio of their investments. If the first friend’s share of the profit is ₹800 less than the second friend’s share, then what is the first friend’s profit?

19. Three friends purchase a bus by investing ₹1,20,000, ₹1,50,000, and ₹1,10,000 respectively. The first friend works as the driver, while the other two work as conductors. They decide that \(\frac{2}{5}\) of the income will be distributed based on work in the ratio 3 : 2 : 2, and the remaining amount will be divided according to their capital investment. If the total income in a certain month is ₹29,260, determine how much each person will receive.

20. Two points on the ground lie along the same straight line with the base of a vertical pillar. From these two points, the angles of elevation to the top of the pillar are complementary. If the distances from the two points to the base of the pillar are 9 meters and 16 meters respectively, and both points are on the same side of the pillar, find the height of the pillar.

21. If the ratio of the heights of two right circular cones is 1:4 and the ratio of their radii is 2:1, then the ratio of their volumes will be ______.

22. The volume of a cone is jointly proportional to the square of the radius of the base and its height. If the ratio of the radii of two cones is 2:3 and their heights are in the ratio 5:4, then find the ratio of their volumes.

23. If the ratio of the radii of two right circular solid cones is 2:3 and the ratio of their heights is 5:3, then what is the ratio of their volumes?

24. If the ratio of the radii of two solid right circular cones is 2:3 and the ratio of their heights is 5:3, then the ratio of their volumes will be 27:30.

25. If the height ratio of two right circular cones is 1:3 and the ratio of their radii is 3:1, then show that the ratio of their volumes is 3:1.

26. A solid hemisphere and a solid cone have the same base diameter and height. Determine the ratio of their volumes and the ratio of their curved surface areas.

27. If two individuals invest \(a\) currency units and \(b\) currency units in a business for \(x\) and \(y\) months respectively, then their profit ratio is -

(a) \(ay:bx\) (b) \(ab:xy\) (c) \(ax:by\) (d) none of the above

28. Prove that if a perpendicular is drawn from the right-angled vertex of any right-angled triangle to the hypotenuse, then the two resulting triangles on either side of the perpendicular are similar to each other and each is also similar to the original triangle.

29. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other and each is similar to the original triangle.

30. Two poles are placed 120 meters apart, and the height of one pole is double that of the other. If the angles of elevation to the tops of the two poles from the midpoint of the line joining their bases are complementary, find the height of the shorter pole.