Question

Shown below are perspective views of a hexagonal prism, a cube, and a cylinder, all having height 10 cm. If the objects are cut by straight planes to generate various cross-sections, which of the statements is/are TRUE?

• A. R can reveal curvilinear cross section; P and R can reveal a square cross section
• B. P, Q, and R can reveal rectangular cross sections; P and Q can reveal isosceles triangle
• C. P and Q can reveal regular hexagonal cross sections; R can reveal a square
• D. P and Q can reveal triangular cross sections

Hint:

For cross-section problems, visualize how each object could be cut at different angles. Circular and symmetrical objects like cylinders may reveal curvilinear sections.

Answer 1

The Correct Option is A, B, C

Step 1: The hexagonal prism (P) when cut can reveal various polygonal cross-sections, including hexagonal and rectangular sections. However, a square cross-section is not possible with this object unless cut in a very specific direction.
Step 2: The cube (Q) can easily reveal square cross-sections as it's a regular 3D object. It can also produce rectangular and other polygonal sections.
Step 3: The cylinder (R), due to its circular symmetry, can reveal curvilinear sections if cut at an angle to its axis. It can also reveal square cross-sections when cut perpendicular to its axis. Hence, both P and R can reveal square cross-sections under certain orientations of the cutting plane.
Step 4: After evaluating all possibilities, we conclude that the correct answers are (A), (B), and (C).

Answer 2

The figure shows a hexagonal prism (P), a cube (Q), and a cylinder (R), each 10 cm tall with a 10 cm width or diameter across the base. The question asks which statement about the possible cross-sections is true when each solid is sliced by a straight cutting plane. Let's check each option against how these three solids actually behave under a straight cut.

1. R can reveal curvilinear cross section; P and R can reveal a square cross section: A cylinder cut at an angle to its axis produces an ellipse, which is a curved (curvilinear) boundary, so the first part is true. For the square claim, since R has both its diameter and height equal to 10 cm, cutting straight through its axis lengthwise gives a rectangle with sides 10 cm by 10 cm, which is a square. The hexagonal prism P, cut straight through two opposite faces along its axis, can also be sized to give a 10 by 10 cm rectangle, which again is a square. Both parts of this option hold up.
2. P, Q, and R can reveal rectangular cross sections; P and Q can reveal isosceles triangle: All three solids can give a rectangle when cut along their axis, but P is a hexagonal prism, and cutting it with a single straight plane does not naturally produce a simple triangle the way slicing off a cube's corner does, so pairing P with Q for triangular sections does not hold up as cleanly.
3. P and Q can reveal regular hexagonal cross sections; R can reveal a square: P's own base is already a regular hexagon, so a cut perpendicular to its axis trivially gives one, but getting an exact regular hexagon from a cube needs a very specific angled cut through the midpoints of six edges, which is a much narrower claim to make in general, so this option overstates how easily Q gives a hexagon.
4. P and Q can reveal triangular cross sections: Q, a cube, can reveal a triangle by slicing off one corner, but this option leaves out the square and curved sections that are clearly visible in the figure's more direct cuts, so it is too narrow to be the best description.

Weighing all four, the first option matches both the curved section from the cylinder and the square section achievable from both the hexagonal prism and the cylinder, using the given 10 cm by 10 cm proportions.

So the correct answer is "R can reveal curvilinear cross section; P and R can reveal a square cross section."