1. 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.)
2. Draw a square having the same area as a rectangle of length 10 cm and width 5 cm. (Only construction marks are required.)
3. Draw a square with the same area as a rectangle whose side lengths are 7.9 cm and 4.1 cm.
4. 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.
5. Draw a square with the same area as an equilateral triangle with a side length of 7 cm.
6. I draw a square having the same area as that of an equilateral triangle with each side measuring 7 cm.
7. Draw a square with the same area as a rectangle whose side lengths are 6 cm and 4 cm.
8. Draw a square with the same area as a rectangle whose side lengths are 4.2 cm and 3.5 cm.
9. 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 \(x\).
10. 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.
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. 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.
13. Draw a triangle with a base length of 6.2 cm and the two adjacent angles measuring 50° and 75°; then construct its incircle. (Only provide the construction marks.)
14. Draw a triangle with side lengths of 5 cm, 6 cm, and 7 cm. Then, draw the incircle of that triangle (only construction marks should be included).
15. 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.)
16. The length of a square-based brass plate is \(x\) cm, with a thickness of 1 millimeter, and the plate weighs 4725 grams. If the weight of 1 cubic centimeter of brass is 8.4 grams, calculate and write the value of \(x\).
17. PQ = 7.5 cm, ∠QPR = 45°, ∠PQR = 75°; PQ = 7.5 cm, ∠QPS = 60°, ∠PQS = 60°; Draw ∆PQR and ∆PQS such that points R and S lie on the same side of PQ. Draw the circumcircle of ∆PQR, and observe the position of point S with respect to this circumcircle — whether it lies inside, on, or outside the circle — and explain your reasoning.
18. Draw a square with the same area as a rectangle whose side lengths are 8 cm and 6 cm.
19. Draw a square with the same area as a triangle whose side lengths are 10 cm, 7 cm, and 5 cm.
20. 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.)
21. 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.)
22. 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.)
23. 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.)
24. Draw the incircle of an equilateral triangle with side length 10 cm. (Only construction marks are required.)
25. 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.)
26. Draw an isosceles triangle with a base length of 5.2 cm and each of the equal sides measuring 7 cm. Then, construct the circumcircle of the triangle and measure the circumradius. (Only mark the construction steps).
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. Draw an isosceles triangle with a base length of 10 cm and an equal angle of 45°. Draw the incircle of the triangle and write the value of the inradius. (Only marking for construction should be included.)
29. 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.
30. Draw a rectangle PQRS where PQ = 4 cm and QR = 6 cm. Draw the two diagonals of the rectangle. Without drawing, calculate and write the position of the circumcenter of ∆PQR and the length of its circumradius. Then, draw the circumcircle of ∆PQR to verify.
