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. 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.

3. 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.

4. 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]

5. 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.

6. 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).

7. 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

8. 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.)

9. 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.)

10. 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.)

11. A right-angled triangle where the hypotenuse is 12 cm and one of the other sides is 5 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]

12. A right-angled triangle in which the hypotenuse is 9 cm and one of the other sides is 5.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

13. "In the right-angled triangle ABC, AC is the hypotenuse. When the triangle is rotated once completely around side AB as the axis, the radius of the generated right circular cone is BC."

14. 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.

15. Draw a right-angled triangle in which the two sides adjacent to the right angle are 5 cm and 6 cm. Then draw a square that has the same area as that triangle.

16. If a triangle similar to one with sides 4 cm, 6 cm, and 8 cm has its largest side measuring 6 cm, what is the length of the smallest side of that triangle?

(a) 4 cm (b) 3 cm (c) 2 cm (d) 5 cm

17. 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.

18. 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.

19. Perfectly explained. The triangle formed is a right-angled isosceles triangle, where the angle of elevation is 45°, making the opposite and adjacent sides equal. So, the height of the pillar equals the distance from its base: 20 meters.

(a) 1:2:3 (b) 1:5:7 (c) 1:7:9 (d) None of the above

20. Draw a right-angled triangle with its adjacent sides measuring 7 cm and 9 cm. Then, draw an incircle for the triangle. (Provide only the construction steps.)

21. Sathi has drawn a right-angled triangle where the length of the hypotenuse is 6 cm more than twice the length of the smallest side. If the length of the third side is 2 cm less than the hypotenuse, then calculate and write the lengths of all three sides of the triangle.

22. 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.

23. In the right-angled triangle ABC, AC is the hypotenuse. When the triangle is rotated once fully around side AB as the axis, the radius of the generated right circular cone is ________.

24. 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.

25. 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.

26. In right-angled triangle \(\triangle ABC\), \(\angle ABC = 90^\circ\), AB = 3 cm, BC = 4 cm. If BD is perpendicular to AC, then find the length of BD.