1. 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.)
2. 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.)
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. 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?
7. Draw a right-angled triangle whose hypotenuse is 10 cm and one of the other sides is 6.5 cm. Then, draw the incircle of this triangle. (Only construction marks are required.)
8. If the diagonal of a rectangle is 10 cm and its area is 62.5 square cm, then the sum of the length and breadth of the rectangle is —
(a) 12 cm (b) 15 cm (c) 20 cm (d) 25 cm
9. 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.
10. The sum of the length, width, and height of a right rectangular prism is 25 cm, and its total surface area is 264 square cm. Determine the length of its diagonal.
11. 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?
12. 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, calculate the length of the pipe.
13. Draw a square with the same area as a rectangle whose side lengths are 8 cm and 6 cm.
14. Draw a square with the same area as a rectangle whose side lengths are 6 cm and 4 cm.
15. Draw a square with the same area as a rectangle whose side lengths are 7.9 cm and 4.1 cm.
16. - sin 280 sec 210 - 1 The dimensions of a cuboid are 12 cm, 6 cm, and 3 cm. What will be the length of each edge of a cube having the same volume as that cuboid?
17. The sum of the length, width, and height of a rectangular box is 24 cm, and the length of its diagonal is 15 cm. What is the total surface area of the rectangular box?
(a) 360 square cm (b) 221 square cm (c) 351quare cm (d) 256 quare cm
18. Let the length of each edge of the cube be \(a\) cm \(\therefore\) The total surface area of the cube = \(6a^2\) square cm If the edge length is increased by 20%, the new edge length = \(a + a \times \cfrac{20}{100}\) cm \(= a + \cfrac{a}{5} = \cfrac{6a}{5}\) cm \(\therefore\) New total surface area of the cube = \(6\left(\cfrac{6a}{5}\right)^2\) square cm = \(\cfrac{216a^2}{25}\) square cm \(\therefore\) Percentage increase in surface area \(= \cfrac{\cfrac{216a^2}{25} - 6a^2}{6a^2} \times 100\%\) \(= \cfrac{(216a^2 - 150a^2) \times 100}{25 \times 6a^2}\%\) \(= \cfrac{4 \times 66a^2}{6a^2}\%\) \(= 44\%\)
19. The diagonal of a cuboid is √725 cm and its volume is 3000 cubic cm. The total surface area of the cuboid is 1300 square cm. Find the length, breadth, and height of the cuboid.
20. Draw a triangle ABC such that BC = 6 cm, CA = 5.5 cm, and AB = 4.5 cm. Then draw the incircle of ∆ABC. (Only construction marks are required.)
21. Draw triangle ABC such that BC = 7 cm, AB = 5 cm, and AC = 6 cm. Then draw the circumcircle of triangle ABC. (Only construction marks are required.)
22. Draw the incircle of an equilateral triangle with side length 10 cm. (Only construction marks are required.)
23. 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.
24. 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
25. A hollow cylindrical iron pipe open at both ends has an inner radius and outer radius of 4 cm and 5 cm respectively. If the total surface area of the pipe is 1188 square centimeters, what is the length of the pipe?
26. Draw a right-angled triangle whose hypotenuse is 9 cm and one of the other sides is 5.5 cm. Then draw an incircle of the triangle. (Only the construction marks are required.)
27. 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.)
28. 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.
29. Draw an isosceles triangle where the base length is 5 cm and each of the equal sides is 6 cm. Then, draw the incircle of the triangle. (Only construction marks should be indicated in each case.)
30. Draw a square with the same area as an equilateral triangle with a side length of 7 cm.
