1. 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.
2. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of a side of the first triangle is 4 cm, then the length of the corresponding side of the second triangle will be ____ .
3. If the volumes of two vertical solid cylinders are equal, and their heights are in the ratio 1:2, then the ratio of the lengths of their radii will be –
(a) 1: √2 (b) √2:1 (c) 1:2 (d) 2:1
4. If the ratio of the areas of two similar triangles is 64:49, then find the ratio of their corresponding sides.
5. 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.
6. 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
7. If the angles of a triangle are in the ratio 1:1:2, then the sides of the triangle will be in the ratio ———.
8. If the ratio of the lengths of two corresponding sides of two similar triangles is 7:11, then their perimeter ratio is _____.
9. “The perimeters of two similar triangles are 24 cm and 16 cm respectively. If one side of the second triangle is 6 cm, what will be the length of the corresponding side of the first triangle.”
10. The corresponding sides of two congruent triangles will be equal.
11. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.
12. The lengths of two sides are 7.6 cm and 6 cm, and the included angle between them is 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).
13. If the heights of two right circular cones are in the ratio 1 : 3 and the lengths of their radii are in the ratio 3 : 1 respectively, then show by calculation that the ratio of their volumes will be 3 : 1.
14. 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.
15. Match the items on the left with those on the right (any two):
16. 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
17. 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
18. If the volumes of two vertical circular cylinders are equal, and their heights are in the ratio 4:9, then the ratio of their radii will be –
19. If the volumes of two solid right circular cylinders are equal and their heights are in the ratio 1:2, what will be the ratio of the lengths of their radii?
(a) 1:√2 (b) √2:1 (c) 1:2 (d) 2:1
20. If two cubes with a side length of \(2\sqrt6\) cm are placed side by side, then the length of the diagonal of the resulting cuboid will be—?
(a) 10 cm (b) 6 cm (c) 2 cm (d) 12 cm
21. If 7, x, y, 189 are in continued proportion, then the values of x and y respectively will be:
(a) 63,21 (b) 21,23 (c) 21,63 (d) 23,21
22. If the radii of two cones are in the ratio 2:3 and their heights are in the ratio 5:3, then what will be the ratio of their volumes?
(a) 4:9 (b) 9:4 (c) 27:20 (d) 20:27
23. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.
24. 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.)
25. 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?
26. If the three sides of a triangle are in the ratio 5 : 12 : 13, then the triangle will always be a right-angled triangle.
27. If \(x, 2x, 3\), and \(y\) are in continued proportion, then what will be the value of \(y\)?
(a) \(4x\) (b) \(6x\) (c) 4 (d) 6
28. If two cones have equal base radii and their heights are in the ratio 2:3, then the ratio of their volumes will be ______.
29. \(\triangle\)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.
30. If the volumes of two cones are in the ratio \(1:4\) and their radii are in the ratio \(4:5\), then what will be the ratio of their heights?
(a) 1:5 (b) 5:4 (c) 5:16 (d) None of the above
