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Difficulty: Hard2D Mental Rotation

You are assembling a square frame using four identical asymmetrical L-shaped brackets. Each bracket consists of a long arm and a short arm meeting at a 90-degree corner. When installed, the right-angle corner of each bracket sits exactly in one of the four corners of the square frame.

The installation requirements are:
- Top-Left Corner: The long arm extends from the corner along the top edge (pointing East), and the short arm extends along the left edge (pointing South).
- Top-Right Corner: The long arm extends from the corner along the right edge (pointing South), and the short arm extends along the top edge (pointing West).
- Bottom-Right Corner: The long arm extends from the corner along the bottom edge (pointing West), and the short arm extends along the right edge (pointing North).
- Bottom-Left Corner: The long arm extends from the corner along the left edge (pointing North), and the short arm extends along the bottom edge (pointing East).

A bracket on your workbench is currently oriented so that its long arm points North-East and its short arm points South-East.

Determine the exact 2D clockwise rotation needed to reorient this workbench bracket for each of the four installation corners. Arrange the corners in order from the LEAST clockwise rotation to the MOST clockwise rotation required (from 0° to 360°).

  1. 1Top-Left Corner
  2. 2Top-Right Corner
  3. 3Bottom-Right Corner
  4. 4Bottom-Left Corner

Answer

The correct order based on clockwise rotation is: Top-Left Corner (45°), Top-Right Corner (135°), Bottom-Right Corner (225°), and Bottom-Left Corner (315°).
By mapping the long arm of the bracket to a compass rose, the starting position is established at 45° (North-East). The target positions for the long arm at each corner are East (90°), South (180°), West (270°), and North (360°). Subtracting the starting 45° from these targets yields clockwise rotations of 45°, 135°, 225°, and 315° respectively.

Step-by-Step Solution

1
Identify the starting orientation of the workbench bracket on a standard compass rose.
The long arm points to North-East (45° clockwise from North) and the short arm points to South-East (135°).
Establishing a baseline angular coordinate allows for precise calculation of rotations.
2
Determine the required rotation for the Top-Left Corner.
Target long arm is East (90°). The rotation from 45° to 90° is a 45° clockwise rotation.
This represents the smallest clockwise adjustment needed among the four corners.
3
Determine the required rotation for the Top-Right Corner.
Target long arm is South (180°). The rotation from 45° to 180° is a 135° clockwise rotation.
Tracking a single reference point (the long arm) ensures consistent and accurate rotation calculations.
4
Determine the required rotation for the Bottom-Right Corner.
Target long arm is West (270°). The rotation from 45° to 270° is a 225° clockwise rotation.
Continuing clockwise through the compass points calculates the third position in the sequence.
5
Determine the required rotation for the Bottom-Left Corner.
Target long arm is North (360° or 0°). The rotation from 45° to 360° is a 315° clockwise rotation.
This represents the largest clockwise rotation, stopping just 45° short of a full circle.

Key Concept

2D Mental Rotation and Directional Tracking
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Topics

2D Mental Rotation

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