Triangle P and pentagon Q have markings on them as shown in the figure. If they are placed over each other, which of the following arrangements is/are possible?
Show Hint
In mental superposition problems, focus on a few key features. Here, the vertices of the shapes and the starting points of the internal lines are the most important features. Check if the configuration in the options preserves these fundamental properties.
Step 1: Understanding the Concept:
This is a spatial reasoning problem that requires mental manipulation (rotation, translation, and superposition) of two geometric shapes, P (a triangle) and Q (a pentagon), each with unique internal markings. We need to determine which of the final arrangements (A, B, C, D) can be formed by overlaying P and Q.
Step 2: Detailed Explanation:
We will analyze each option by trying to find an orientation for both P and Q that results in the given configuration. The key features to track are the positions of the vertices of the shapes and the relative positions of the markings (the line with a circle and the line with a cross).
Option A: This arrangement is possible. The pentagon Q is placed in its standard orientation. The triangle P is rotated by approximately 180 degrees and placed over the pentagon such that one of its vertices points downwards. The internal markings align as shown.
Option B: This arrangement is possible. The pentagon Q is rotated slightly clockwise. The triangle P is rotated significantly clockwise (or flipped and rotated) and placed such that one of its vertices points to the lower left. The markings can be aligned as shown in the figure.
Option C: This arrangement is not possible. In the original shapes, the marking in triangle P is a line from a vertex to the center, and the marking in pentagon Q is a line from a vertex to the center. In option C, the marking with the circle (from P) originates from a side of the triangle, not a vertex. Similarly, the marking with the square (from Q) originates from a side of the pentagon. This contradicts the construction of the original shapes P and Q. Therefore, configuration C cannot be achieved.
Option D: This arrangement is possible. The pentagon Q is rotated so one of its vertices points to the upper right. The triangle P is rotated so one of its vertices points to the upper left. The shapes are then superimposed. The markings can be arranged as shown.
Step 3: Final Answer:
Based on the analysis, the arrangements shown in options A, B, and D are possible through rotation and superposition of the original shapes P and Q, while option C is impossible due to the incorrect origin points of the internal markings.
