1. 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.
2. Draw a square with the same area as an equilateral triangle whose side length is 6 cm.
3. Draw a square with the same area as an equilateral triangle with a side length of 7 cm.
4. Draw a square with the same area as a triangle whose side lengths are 10 cm, 7 cm, and 5 cm.
5. I draw a square having the same area as that of an equilateral triangle with each side measuring 7 cm.
6. Draw a square with the same area as a rectangle whose side lengths are 7.9 cm and 4.1 cm.
7. Draw a square with the same area as a rectangle whose side lengths are 8 cm and 6 cm.
8. Draw a square with the same area as a rectangle whose side lengths are 6 cm and 4 cm.
9. Draw a square with the same area as a rectangle whose side lengths are 4.2 cm and 3.5 cm.
10. 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).
11. There is a square-shaped park in our neighborhood. A rectangular park, whose length is 5 meters more and breadth is 3 meters less than the side of this square park, has an area that is 78 square meters less than twice the area of the square park. Find and write the length of the side of the square park.
12. Draw an equilateral triangle with side length 7 cm. Then draw both the circumcircle and the incircle of the triangle. Using a scale, determine the lengths of the circumradius and the inradius. Also, write whether there is any relationship between them.
13. Draw an isosceles triangle whose base is 7.8 cm and the length of each of the equal sides is 6.5 cm. Then draw an incircle (an inscribed circle) inside that triangle.
14. Draw an equilateral triangle with each side measuring 7 cm. Then, draw the incircle (inscribed circle) of that triangle.
15. 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.)
16. 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
17. "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
18. 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 \(x\).
19. 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?
20. 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.
21. Here is the drawing of an isosceles triangle with a base length of 5.6 cm and equal sides of 7.5 cm each, along with its incircle.
22. An isosceles triangle where the base is 7.8 cm and each of the equal sides is 6.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).
23. 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.
24. Draw an equilateral triangle ABC with each side measuring 5 cm. Then, draw the circumcircle of that triangle. At point A on the circle, draw a tangent. On the tangent, take a point P such that AP = 5 cm. From point P, draw another tangent to the circle, and write down which point on the circle this second tangent touches.
25. 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.
26. 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.
27. 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
28. 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.
29. 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?
30. If the area of the square drawn on one side of any triangle is equal to the sum of the areas of the squares drawn on the other two sides, then prove that the angle opposite to the first side is a right angle.
