1. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm. If the triangle is rotated once completely about the longer of these two sides as the axis, the solid formed is a cone. Calculate and write the lateral surface area, total surface area, and volume of the cone formed.

2. A right-angled triangle has two sides adjacent to the right angle measuring 4 cm and 3 cm. If the triangle is rotated once completely around its hypotenuse as the axis, find the volume of the solid formed.

3. Amina has drawn a right-angled triangle with the two sides adjacent to the right angle measuring 15 cm and 20 cm. When the triangle is revolved once completely around the 15 cm side as the axis, it forms a solid. Calculate the lateral surface area, the total surface area, and the volume of the solid formed.

4. 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.

5. Draw a right-angled triangle whose two sides adjacent to the right angle are 7 cm and 9 cm respectively. Then draw the incircle of that triangle. (Only construction marks are required.)

6. Draw a right-angled triangle whose two sides adjacent to the right angle are 4.5 cm and 6 cm. Then draw the incircle of that triangle. (Only construction marks are required.)

7. In a right-angled triangle, the hypotenuse is 15 cm, and the difference between the other two sides is 3 cm. Find the lengths of those two sides.

8. A right-angled triangle where the two sides adjacent to the right angle are 4 cm and 4 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

9. Draw a right-angled triangle whose two sides adjacent to the right angle are 4 cm and 5 cm. Then draw a circumcircle of that triangle.

10. A right-angled triangular prism has a base area of 100 square centimeters, and the areas of the two adjacent faces to the base are 40 square centimeters and 160 square centimeters. Its volume will be—

(a) 800 cubic cm (b) 880 cubic cm (c) 890 cubic cm (d) 900 cubic cm

11. Draw a right-angled triangle whose two sides adjacent to the right angle are 8 cm and 6 cm respectively, and draw an incircle of the triangle. (Only construction marks are required)

12. The side lengths of two right-angled trapezoids are 4 cm, 12 cm, 15 cm, 6 cm, (2h - 1) cm, and 16 cm, respectively. If the areas of the two trapezoids are equal, find the value of h.

13. A right-angled triangle in which the two sides adjacent to the right angle are 7 cm and 9 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

14. A solid sphere has a total surface area of 1386 cm². It is melted and reshaped into several right circular cones, each with a radius of 3.5 cm and a height of 6 cm. Find the radius of the solid sphere and determine how many such cones are formed.

15. Construct a right-angled triangle in which the two arms adjacent to the right angle are 7 cm and 9 cm. Draw an incircle (inscribed circle) of that triangle and measure its radius. (Each construction step must be marked.)

16. Draw a right-angled triangle with the two sides adjacent to the right angle measuring 4 cm each. Then draw the circumcircle of the triangle. (Only construction marks are required.)

17. 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.

18. The base of a triangle is \(16\sqrt{3}\) cm, and the two angles adjacent to the base are 30° and 60°. What is the height of the triangle?

19. In right-angled triangle ABC, ∠ABC = 90°, AB = 3 cm, BC = 4 cm, and from point B, a perpendicular BD is drawn to side AC, meeting AC at point D. Find the length of BD.

20. Find the radius of the base of a right circular cone whose slant height is 7 cm and total surface area is 147.84 cm².

21. A solid cuboid has a length, width, and height ratio of \(4:3:2\), and its total surface area is \(468\) square cm. Determine the volume of the cuboid.

22. Find the radius of the base of a right circular cone if its slant height is 7 cm and its total surface area is 147.84 cm².

23. The slant height of a right circular cone is 7 cm, and its total surface area is 147.84 square cm. Find the radius of its base.

24. The slant height of a right circular cone is 7 cm and its total surface area is 147.84 square cm. Find the radius of the base and the area of the base of the cone. Let me know if you'd like the full solution worked out as well. I'm happy to help.

25. The radius of the first sphere \( = \cfrac{21}{2} \) cm And the radius of the second sphere \( = \cfrac{17.5}{2} \) cm \( = \cfrac{175}{2 \times 10} \) cm \( = \cfrac{35}{4} \) cm

The ratio of the amount of metal used to make the two spheres = Ratio of their surface areas \( = 4π\left(\cfrac{21}{2}\right)^2 : 4π\left(\cfrac{35}{4}\right)^2 = \cfrac{21 \times 21}{4} : \cfrac{35 \times 35}{16} \) \( = 9 : \cfrac{25}{4} = 36 : 25 \) (Answer)

26. The slant height of a right circular cone is 7 cm and its total surface area is 147.84 sq. cm. Find the radius of the base of the cone.

27. If the lengths of the three sides of a triangle are 4 cm, 15 cm, and 17 cm, then calculate and write whether the triangle is a right-angled triangle or not.

28. In the right-angled triangle ABC, ∠ABC = 90°, AB = 3 cm, BC = 4 cm, and from point B a perpendicular BD is drawn to side AC, meeting it at point D. Find the length of BD.

29. 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.

30. If the slant height of a right circular cone is 7 cm and its total surface area is 147.84 square cm, then find the diameter of the cone.