Figure P shows the top view of a circular table top with few objects arranged on it for a sketching class. Numbers 1 to 24 in figure P represent different directions around the table from which students are sketching the objects. Which option gives the directions from which the sketches in Figure Q and R are drawn?
Show Hint
In perspective problems, identify the object closest to the viewer in the sketch (the one that appears largest or lowest in the frame). Then, locate that object in the top-down view and determine the direction from which it would appear in the foreground.
Step 1: Understanding the Concept:
This problem requires spatial reasoning and the ability to correlate a 2D top-down view with 3D perspective sketches. We need to analyze the relative positions of the objects in the sketches (Q and R) and find the corresponding viewpoint numbers on the circular table (P).
Step 2: Detailed Explanation:
Let's analyze the arrangement of objects on the table from the top view (P). There is a central pyramid, surrounded by cylinders, cones, and cubes.
Analyzing Sketch Q:
Foreground: The sketch shows a tall cylinder in the immediate foreground, slightly to the left.
Middle ground: To the right of the cylinder is a cone, and behind both is the central pyramid. Another shorter cylinder is visible to the far right.
Arrangement: The viewer sees the tall cylinder, then the cone, then the pyramid in a line going away from them towards the right.
Matching to P: Looking at the top view P, this arrangement corresponds to viewing from position 1. From this viewpoint, the tall cylinder at the top is closest, the cone is just behind and to its right, and the pyramid is in the center. The other visible objects match this perspective.
Analyzing Sketch R:
Foreground: The sketch shows a cube in the immediate foreground.
Middle ground: Behind the cube, there are two cones, and behind them is the central pyramid. To the left and right are cylinders.
Arrangement: The cube is directly in front of the viewer. The two cones are positioned behind it, flanking the central pyramid.
Matching to P: This view corresponds to looking from the bottom of the diagram. Specifically, from position 16, a student would see the cube at the bottom edge of the table as the closest object. The two cones and the pyramid would be visible behind it, arranged as shown in the sketch.
Step 3: Final Answer:
The sketch in Figure Q is drawn from direction 1, and the sketch in Figure R is drawn from direction 16. This corresponds to option (C).
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Approach Solution -2
Each option proposes a different pair of viewpoints for the two sketches, so the fastest way to confirm the right pair is to check every option against the one clue each sketch gives away for free, namely which object sits closest to the viewer and what lies immediately behind it.
Q:19, R:5: Position 19 lies on the side of the table associated with the shorter cylinder and cone cluster, not the tall cylinder that Sketch Q places right in front of the viewer, so the foreground object would not match. Position 5 puts a cylinder or cone nearest the viewer rather than the cube that dominates the foreground of Sketch R, so this pair fails on both counts.
Q:8, R:15: From position 8 the nearest object works out to be a cone rather than the tall cylinder Sketch Q needs in front, and position 15 does not bring the cube close enough to the edge of the table to produce the large, near foreground cube seen in Sketch R.
Q:1, R:16: From position 1, the tall cylinder is the closest object, with the cone just behind and to its right and the pyramid centred further back, exactly as Sketch Q shows. From position 16, the cube sits right at the near edge, with the two cones and the pyramid receding behind it, matching Sketch R feature for feature.
Q:24, R:15: Position 24 does not place the tall cylinder as the nearest object, and position 15, as already noted, does not bring the cube close enough to the viewer to reproduce Sketch R.
Only the pair Q:1, R:16 survives this check against both sketches at once, so the sketches were drawn from directions 1 and 16.
