1. If two cubes, each with side length \(2\sqrt{6}\), are placed side by side, the resulting cuboid will have a diagonal of –

(a) 10cm (b) 12 cm (c) 6 cm (d) 8cm

2. Two cubes with equal edge lengths are joined side by side to form a cuboid. If the volume of the cuboid is 1024 cubic centimeters, what is the length of each edge of the cubes?

3. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of a side of the first triangle is 4 cm, then the length of the corresponding side of the second triangle will be ____ .

4. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.

5. If the lengths of the sides of two triangles are in proportion, then the triangles will be ——.

6. 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 \(x\).

7. In a circle with a radius of 5 cm, AB and AC are two equal chords. The center of the circle is located outside the triangle ABC. If AB = AC = 6 cm, determine the length of the chord BC.

8. 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.

9. “The perimeters of two similar triangles are 24 cm and 16 cm respectively. If one side of the second triangle is 6 cm, what will be the length of the corresponding side of the first triangle.”

10. In a circle with a radius of 5 cm, AB and AC are two equal-length chords. The center of the circle lies outside triangle ABC. Given AB = AC = 6 cm, find the length of chord BC.

11. If the diameter of a right circular cylinder is 3 cm and its height is 4 cm, then the longest rod that can be placed inside the cylinder will have a length of ___________ cm.

12. Draw a triangle where one side is 6.7 cm and the two adjacent angles are 75° and 55°. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]

13. The lengths of two sides are 7.6 cm and 6 cm, and the included angle between them is 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

14. The length of one side is 6.2 cm, and the measures of the two angles adjacent to that side are 50° and 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

15. 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).

16. In our neighborhood, there are two houses on opposite sides of the road facing each other. If the base of a ladder is placed 6 meters away from the foot of the wall of the first house and the ladder is leaned against the wall, it makes an angle of 30° with the horizontal. But if the same ladder is placed in the same position and leaned against the wall of the second house, it makes an angle of 60° with the horizontal. (i) Find the length of the ladder. (ii) Calculate how far the foot of the ladder is from the base of the second house. (iii) Find the width of the road. (iv) Determine the height at which the top of the ladder touches the second house.

17. "If one of the diagonals of a rhombus with a side length of 10 cm is 12 cm, then calculate and write the length of the other diagonal."

18. The perimeters of two similar triangles are 27 cm and 16 cm respectively. If one side of the first triangle is 9 cm, then find the length of the corresponding side of the second triangle.

19. If three metallic cubes with edges measuring 3 cm, 4 cm, and 5 cm are melted and recast into a new cube, the length of the edge of the new cube will be –

(a) 6 cm (b) 7 cm (c) 8 cm (d) 9 cm

20. If one diagonal of a rhombus with side length 17 cm is 16 cm, what is the length of the other diagonal?

(a) 30 cm (b) 25 cm (c) 32 cm (d) 18 cm

21. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

22. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

23. Two circles with centers A and B have a common tangent PO of length 16 cm, and the length of PB is 10 cm; the area of triangle \(\triangle\)POB will be—

(a) 24 square cm (b) 48 square cm (c) 16 square cm (d) 12 square cm

24. The radius of the circumcircle of an equilateral triangle with side length \(5\sqrt{3}\) cm will be —

(a) 6 cm (b) 5 cm (c) 4 cm (d) 3 cm

25. 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.

26. If a right circular cylinder has a radius of 3 cm and a height of 4 cm, then what is the length of the longest rod that can be placed inside it?

27. The length, width, and height of a cuboidal room are respectively \(a\), \(b\), and \(c\) units, and given that \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), what is the length of the longest rod that can be placed inside the room?

28. Draw the incircle of an equilateral triangle with side length 10 cm. (Only construction marks are required.)

29. A rectangular wooden box has the sum of its dimensions equal to 10 cm. The longest rod that can be placed inside it has a length of \(\sqrt{38}\) cm. Find the total surface area of the box.

30. A rod of length \(6\sqrt{3}\) cm can be placed inside a cube along its longest diagonal. What is the volume of the cube?