Determine if an inner square can touch an outer square at 4 points only when the inner square is at specific rotation angles | Step-by-Step Solution

What We're Solving:

We need to figure out whether an inner square can only touch an outer square at exactly 4 points when it's positioned in a specific way, and how rotating the inner square changes where these contact points occur.

The Approach:

This is a fascinating geometry problem that combines coordinate geometry, rotation, and spatial reasoning! We'll use mathematical modeling to explore different scenarios. The key insight is that we need to think about what "touching at 4 points" actually means - it suggests the inner square's corners are touching the outer square's sides.

Step-by-Step Solution:

Step 1: Set up our coordinate system Let's place the outer square with vertices at (±1, ±1) for simplicity. This gives us sides along x = ±1 and y = ±1.

Step 2: Understand the "4 contact points" condition For the inner square to touch the outer square at exactly 4 points, the most logical scenario is that each vertex of the inner square touches one side of the outer square. This creates a very specific geometric constraint!

Step 3: Analyze the "halfway point" claim Let's start with the inner square rotated 45° relative to the outer square. If the inner square's vertices touch the midpoints of the outer square's sides, the contact points would be at (1,0), (0,1), (-1,0), and (0,-1).

For this to work, we need to find what size the inner square should be. Using the distance formula, if the inner square is centered at the origin and rotated 45°, its vertices are at (±d, 0) and (0, ±d) where d is half the diagonal length.

Step 4: Derive the relationship When the inner square touches at the midpoints, we have d = 1. The side length of the inner square would be d√2 = √2. This means the inner square has half the side length of the outer square (since the outer square has side length 2).

Step 5: Explore other rotation angles Here's where it gets interesting! Let's say we rotate the inner square by angle θ from the 45° position. The vertices will trace out circles, and they'll still touch the outer square's sides, but at different points - not necessarily the midpoints!

Step 6: Test the general case Through mathematical analysis (involving trigonometry), we can show that for any rotation angle, there exists exactly one size of inner square that will have all 4 vertices touching the outer square's sides. The contact points move along the sides as we rotate!

The Answer:

No, the inner square does NOT need to touch at the halfway points of the outer square's sides! The halfway point contact only occurs at specific rotation angles (45°, 135°, 225°, 315° relative to the outer square). For other rotation angles, the contact points slide along the sides of the outer square, but the 4-point contact condition can still be satisfied by adjusting the size of the inner square.

Memory Tip:

Think of it like a coin rolling inside a square box - the coin can touch all four sides at once in many different orientations, not just one special position! The "halfway point" is just one special case in a family of solutions.

Great question! This problem beautifully demonstrates how geometric constraints can have multiple solutions, and how rotation creates a continuous family of configurations. Keep exploring these spatial relationships - they're the foundation of advanced geometry and engineering!