Question

A solid object with triangular groove is shown on the left. Which option(s) along with this object can form the combination shown on the right?

• A. Fig A
• B. Fig B
• C. Fig C
• D. Fig D

Hint:

When an inclined cut intersects a regular orthogonal block at an angle, it creates wedge-shaped pockets. Conversely, filling that wedge pocket to restore orthogonal steps always requires a standard rectangular block.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are presented with a green triangular prism (wedge) featuring a recessed groove cut into its inclined face. On the right, we see a composite assembly where a white solid block fills this groove and protrudes outward to create a level horizontal step. We must identify the exact 3D geometry of the white block from the wireframe options.

Step 2: Key Formula or Approach:
We solve this by performing boolean solid geometry addition. The white block is the exact union of two distinct volumetric regions:
\[ V_{\text{white}} = V_{\text{internal}} \cup V_{\text{external}} \]

• \(V_{\text{internal}}\) must be identical to the negative volume of the pocket cut into the green wedge.

• \(V_{\text{external}}\) is the visible volume projecting outside the inclined plane.


Step 3: Detailed Explanation:

• Let us analyze the interior geometry (\(V_{\text{internal}}\)):
- The dashed lines in the left image reveal that the groove has a vertical back wall, a horizontal bottom wall, and vertical side walls.
- The deepest interior corner where the back and bottom walls meet forms a mutually perpendicular 90-degree trihedral vertex.
- Therefore, the back half of the white block must possess mutually perpendicular horizontal and vertical faces.

• Let us analyze the exterior geometry (\(V_{\text{external}}\)):
- The combined drawing shows that the white block extends outward, terminating in a flat horizontal top surface and a flat vertical front surface.

• Merging these two regions across the shared inclined boundary plane eliminates the slope entirely. Every single face of the completed white block is either horizontal or vertical.

• A six-sided regular polyhedron whose faces are all mutually orthogonal is a rectangular cuboid (or a cube).

• Let us inspect the options:
-

Option A: Represents a triangular wedge wireframe. Inserting this into the pocket would simply make the inclined face flush, failing to create the protruding horizontal platform.
-

Option B: Represents a standard cube wireframe. Sinking one corner of this cube into the orthogonal pocket leaves the opposite corner protruding outside, forming the exact horizontal platform shown in the problem.
-

Option C: Represents a cylinder quadrant featuring a curved face.
-

Option D: Represents a notched stepped cuboid.


Step 4: Final Answer:
Option (B) is the correct block required to form the combination.