1. Draw a square with the same area as a rectangle whose side lengths are 8 cm and 6 cm.
2. Draw a square with the same area as a rectangle whose side lengths are 6 cm and 4 cm.
3. Draw a square with the same area as a rectangle whose side lengths are 4.2 cm and 3.5 cm.
4. Draw a square with the same area as a triangle whose side lengths are 10 cm, 7 cm, and 5 cm.
5. Draw a square with the same area as an isosceles triangle whose base length is 7 cm and each of the equal sides is 5 cm.
6. Draw a rectangle with sides of 8 cm and 6 cm, and then draw a square having the same area as that rectangle. (Only construction marks are required.)
7. Draw a square with the same area as an equilateral triangle whose side length is 6 cm.
8. Draw a square with the same area as an equilateral triangle whose side length is 7 cm.
9. Draw a square having the same area as a rectangle of length 10 cm and width 5 cm. (Only construction marks are required.)
10. Draw a rectangle with a length of 7 cm and a width of 4 cm; now draw a square with the same area as that rectangle. (Only the construction marks are required.)
11. Draw a square with the same area as an equilateral triangle with a side length of 7 cm.
12. I draw a square having the same area as that of an equilateral triangle with each side measuring 7 cm.
13. 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.
14. The lengths of the two equal sides of an isosceles triangle are 5 cm, and the base is 6 cm. The area of the triangle is —
(a) 18 square cm (b) 12 square cm (c) 15 square cm (d) 30 square cm
15. Determine the radius of the base of a solid right circular cone with a slant height of 7 cm and a total surface area of 147.84 square cm.
16. 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.
17. 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.
18. The radius of a circle with center O is 5 cm. From an external point P, which is located at a certain distance from point O, two tangents PQ and PR are drawn to the circle. The quadrilateral PQOR has an area of 60 square centimeters. Find the distance from point O to point P.
19. Two solid spheres with radii of 1 cm and 6 cm are melted and recast into a hollow sphere with a thickness of 1 cm. Find the outer radius of the new hollow sphere and calculate its curved surface area.
20. A hollow vertical cylindrical iron pipe has an outer radius of 5 cm and an inner radius of 4 cm. If the total surface area of the pipe is 1188 square cm, what is the length of the pipe?
21. A solid sphere has a surface area of 616 square cm. It is melted to form 14 identical right circular cones, each with a height of 2 cm. Find the diameter of the base of each cone.
22. A conical-shaped head cap made of thermocol has a base diameter of 21 cm. Its upper surface is covered with foil, and the cost of covering is 10 paise per square centimeter, totaling ₹57.75. Find the height and slant height of the cap.
23. "The differences between the semi-perimeter of a triangle and the lengths of its sides are respectively 4 cm, 7 cm, and 5 cm. The area of the triangle is –"
(a) \(20\sqrt{7}\) square cm (b) \(10\sqrt{14}\) square cm (c) \(20\sqrt{14}\) square cm (d) 140 square cm
24. The area of a trapezium-shaped field is 162 square centimeters and its height is 6 cm. If one of the parallel sides of the trapezium is 23 cm long, then the length of the other parallel side is –
(a) 29 cm (b) 31 cm (c) 32 cm (d) 33 cm
25. A brass plate has a square base with side length \(x\) cm, thickness 1 mm, and a total weight of 4725 grams. If the weight of 1 cubic centimeter of brass is 8.4 grams, then calculate and write the value of \(x\).
26. A rectangular container has a base in the shape of a rectangle with a length of 60 cm and a width of 45 cm. The container has a height of 20 cm and is half-filled with water. Determine the side length of a metallic cube that, when placed in the container, will cause the water level to reach the brim.
27. 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.
28. The outer dimensions of a box with a lid are 12 cm in length, 10 cm in width, and 8 cm in height. The total inner surface area of the box is 376 square cm. If the walls of the box have equal thickness, what is their thickness?
29. 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.
30. Draw a triangle with side lengths of 7 cm, 6 cm, and 5.5 cm. Draw the incircle of the triangle and measure the radius of the incircle.
