Diagonal Fold - Basic: Worksheet 2 - Beginner Practice Diagonal Fold - Basic BEGINNER

Detailed Step-by-Step Solution:

Step 1 - Understanding Diagonal Folds:
- Diagonal folds create 45-degree symmetry
- The fold line runs corner to corner (top-left to bottom-right)
- Creates a triangular shape with 2 layers
- Fold direction: diagonally from top-left to bottom-right

Step 2 - Geometric Analysis:
- Original: Square with 4 corners
- After fold: Triangle with 3 visible corners
- Hidden corner: Bottom-right under folded layers
- Symmetry axis: Diagonal line at 45° from top-left to bottom-right
- Layer structure: 2 overlapping triangular layers

Step 3 - Hole Punch Location:
- Position: near the center of folded triangle
- Coordinate mapping: (45,55) in 100x100 coordinate system
- Penetration: Through both triangular layers
- Important: Position relative to diagonal symmetry axis

Step 4 - Mental Unfolding Technique:
- Keep one triangular layer fixed (bottom layer)
- Rotate the other layer 180° around diagonal axis
- The hole on the moving layer traces to its symmetric position
- Use coordinate transformation: (x,y) → (y,x) for this diagonal
- Result: Two holes diagonally symmetric across the diagonal

Step 5 - Pattern Recognition:
- Diagonal symmetry creates diagonal hole pattern
- Both holes equidistant from diagonal fold line
- Perpendicular distance from fold is equal for both holes
- Visual pattern: Two holes symmetric about the main diagonal
- Final answer: Two holes diagonally symmetric across the diagonal

Pro Visualization Tip:
Imagine the paper as a book cover. When you open it (unfold), the mark on one side appears mirrored on the other side across the spine (fold line).

Advanced Technique:
For diagonal folds, use coordinate geometry. If the hole is at position (x, y) on the folded triangle, the symmetric hole appears at position (y, x) when unfolded across the diagonal.

Common Pitfall: Don't confuse diagonal symmetry with horizontal/vertical symmetry. Diagonal folds at 45° create patterns along that diagonal axis, not along the edges.

Coordinate Verification:
Original hole: (45,55)
Mirror hole: (55,45) [for top-left to bottom-right diagonal]
Check symmetry: Both equidistant from line y=x

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Understanding Diagonal Folds:
- Diagonal folds create 45-degree symmetry
- The fold line runs corner to corner (top-left to bottom-right)
- Creates a triangular shape with 2 layers
- Fold direction: diagonally from top-left to bottom-right

Step 2 - Geometric Analysis:
- Original: Square with 4 corners
- After fold: Triangle with 3 visible corners
- Hidden corner: Bottom-right under folded layers
- Symmetry axis: Diagonal line at 45° from top-left to bottom-right
- Layer structure: 2 overlapping triangular layers

Step 3 - Hole Punch Location:
- Position: near the center of folded triangle
- Coordinate mapping: (45,55) in 100x100 coordinate system
- Penetration: Through both triangular layers
- Important: Position relative to diagonal symmetry axis

Step 4 - Mental Unfolding Technique:
- Keep one triangular layer fixed (bottom layer)
- Rotate the other layer 180° around diagonal axis
- The hole on the moving layer traces to its symmetric position
- Use coordinate transformation: (x,y) → (y,x) for this diagonal
- Result: Two holes diagonally symmetric across the diagonal

Step 5 - Pattern Recognition:
- Diagonal symmetry creates diagonal hole pattern
- Both holes equidistant from diagonal fold line
- Perpendicular distance from fold is equal for both holes
- Visual pattern: Two holes symmetric about the main diagonal
- Final answer: Two holes diagonally symmetric across the diagonal

Pro Visualization Tip:
Imagine the paper as a book cover. When you open it (unfold), the mark on one side appears mirrored on the other side across the spine (fold line).

Advanced Technique:
For diagonal folds, use coordinate geometry. If the hole is at position (x, y) on the folded triangle, the symmetric hole appears at position (y, x) when unfolded across the diagonal.

Common Pitfall: Don't confuse diagonal symmetry with horizontal/vertical symmetry. Diagonal folds at 45° create patterns along that diagonal axis, not along the edges.

Coordinate Verification:
Original hole: (45,55)
Mirror hole: (55,45) [for top-left to bottom-right diagonal]
Check symmetry: Both equidistant from line y=x

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Understanding Diagonal Folds:
- Diagonal folds create 45-degree symmetry
- The fold line runs corner to corner (top-left to bottom-right)
- Creates a triangular shape with 2 layers
- Fold direction: diagonally from top-left to bottom-right

Step 2 - Geometric Analysis:
- Original: Square with 4 corners
- After fold: Triangle with 3 visible corners
- Hidden corner: Bottom-right under folded layers
- Symmetry axis: Diagonal line at 45° from top-left to bottom-right
- Layer structure: 2 overlapping triangular layers

Step 3 - Hole Punch Location:
- Position: near the center of folded triangle
- Coordinate mapping: (45,55) in 100x100 coordinate system
- Penetration: Through both triangular layers
- Important: Position relative to diagonal symmetry axis

Step 4 - Mental Unfolding Technique:
- Keep one triangular layer fixed (bottom layer)
- Rotate the other layer 180° around diagonal axis
- The hole on the moving layer traces to its symmetric position
- Use coordinate transformation: (x,y) → (y,x) for this diagonal
- Result: Two holes diagonally symmetric across the diagonal

Step 5 - Pattern Recognition:
- Diagonal symmetry creates diagonal hole pattern
- Both holes equidistant from diagonal fold line
- Perpendicular distance from fold is equal for both holes
- Visual pattern: Two holes symmetric about the main diagonal
- Final answer: Two holes diagonally symmetric across the diagonal

Pro Visualization Tip:
Imagine the paper as a book cover. When you open it (unfold), the mark on one side appears mirrored on the other side across the spine (fold line).

Advanced Technique:
For diagonal folds, use coordinate geometry. If the hole is at position (x, y) on the folded triangle, the symmetric hole appears at position (y, x) when unfolded across the diagonal.

Common Pitfall: Don't confuse diagonal symmetry with horizontal/vertical symmetry. Diagonal folds at 45° create patterns along that diagonal axis, not along the edges.

Coordinate Verification:
Original hole: (45,55)
Mirror hole: (55,45) [for top-left to bottom-right diagonal]
Check symmetry: Both equidistant from line y=x

Detailed Step-by-Step Solution:

Step 1 - Understanding Diagonal Folds:
- Diagonal folds create 45-degree symmetry
- The fold line runs corner to corner (top-left to bottom-right)
- Creates a triangular shape with 2 layers
- Fold direction: diagonally from top-left to bottom-right

Step 2 - Geometric Analysis:
- Original: Square with 4 corners
- After fold: Triangle with 3 visible corners
- Hidden corner: Bottom-right under folded layers
- Symmetry axis: Diagonal line at 45° from top-left to bottom-right
- Layer structure: 2 overlapping triangular layers

Step 3 - Hole Punch Location:
- Position: near the center of folded triangle
- Coordinate mapping: (45,55) in 100x100 coordinate system
- Penetration: Through both triangular layers
- Important: Position relative to diagonal symmetry axis

Step 4 - Mental Unfolding Technique:
- Keep one triangular layer fixed (bottom layer)
- Rotate the other layer 180° around diagonal axis
- The hole on the moving layer traces to its symmetric position
- Use coordinate transformation: (x,y) → (y,x) for this diagonal
- Result: Two holes diagonally symmetric across the diagonal

Step 5 - Pattern Recognition:
- Diagonal symmetry creates diagonal hole pattern
- Both holes equidistant from diagonal fold line
- Perpendicular distance from fold is equal for both holes
- Visual pattern: Two holes symmetric about the main diagonal
- Final answer: Two holes diagonally symmetric across the diagonal

Pro Visualization Tip:
Imagine the paper as a book cover. When you open it (unfold), the mark on one side appears mirrored on the other side across the spine (fold line).

Advanced Technique:
For diagonal folds, use coordinate geometry. If the hole is at position (x, y) on the folded triangle, the symmetric hole appears at position (y, x) when unfolded across the diagonal.

Common Pitfall: Don't confuse diagonal symmetry with horizontal/vertical symmetry. Diagonal folds at 45° create patterns along that diagonal axis, not along the edges.

Coordinate Verification:
Original hole: (45,55)
Mirror hole: (55,45) [for top-left to bottom-right diagonal]
Check symmetry: Both equidistant from line y=x

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Understanding Diagonal Folds:
- Diagonal folds create 45-degree symmetry
- The fold line runs corner to corner (top-left to bottom-right)
- Creates a triangular shape with 2 layers
- Fold direction: diagonally from top-left to bottom-right

Step 2 - Geometric Analysis:
- Original: Square with 4 corners
- After fold: Triangle with 3 visible corners
- Hidden corner: Bottom-right under folded layers
- Symmetry axis: Diagonal line at 45° from top-left to bottom-right
- Layer structure: 2 overlapping triangular layers

Step 3 - Hole Punch Location:
- Position: near the center of folded triangle
- Coordinate mapping: (45,55) in 100x100 coordinate system
- Penetration: Through both triangular layers
- Important: Position relative to diagonal symmetry axis

Step 4 - Mental Unfolding Technique:
- Keep one triangular layer fixed (bottom layer)
- Rotate the other layer 180° around diagonal axis
- The hole on the moving layer traces to its symmetric position
- Use coordinate transformation: (x,y) → (y,x) for this diagonal
- Result: Two holes diagonally symmetric across the diagonal

Step 5 - Pattern Recognition:
- Diagonal symmetry creates diagonal hole pattern
- Both holes equidistant from diagonal fold line
- Perpendicular distance from fold is equal for both holes
- Visual pattern: Two holes symmetric about the main diagonal
- Final answer: Two holes diagonally symmetric across the diagonal

Pro Visualization Tip:
Imagine the paper as a book cover. When you open it (unfold), the mark on one side appears mirrored on the other side across the spine (fold line).

Advanced Technique:
For diagonal folds, use coordinate geometry. If the hole is at position (x, y) on the folded triangle, the symmetric hole appears at position (y, x) when unfolded across the diagonal.

Common Pitfall: Don't confuse diagonal symmetry with horizontal/vertical symmetry. Diagonal folds at 45° create patterns along that diagonal axis, not along the edges.

Coordinate Verification:
Original hole: (45,55)
Mirror hole: (55,45) [for top-left to bottom-right diagonal]
Check symmetry: Both equidistant from line y=x

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Understanding Diagonal Folds:
- Diagonal folds create 45-degree symmetry
- The fold line runs corner to corner (top-left to bottom-right)
- Creates a triangular shape with 2 layers
- Fold direction: diagonally from top-left to bottom-right

Step 2 - Geometric Analysis:
- Original: Square with 4 corners
- After fold: Triangle with 3 visible corners
- Hidden corner: Bottom-right under folded layers
- Symmetry axis: Diagonal line at 45° from top-left to bottom-right
- Layer structure: 2 overlapping triangular layers

Step 3 - Hole Punch Location:
- Position: near the center of folded triangle
- Coordinate mapping: (45,55) in 100x100 coordinate system
- Penetration: Through both triangular layers
- Important: Position relative to diagonal symmetry axis

Step 4 - Mental Unfolding Technique:
- Keep one triangular layer fixed (bottom layer)
- Rotate the other layer 180° around diagonal axis
- The hole on the moving layer traces to its symmetric position
- Use coordinate transformation: (x,y) → (y,x) for this diagonal
- Result: Two holes diagonally symmetric across the diagonal

Step 5 - Pattern Recognition:
- Diagonal symmetry creates diagonal hole pattern
- Both holes equidistant from diagonal fold line
- Perpendicular distance from fold is equal for both holes
- Visual pattern: Two holes symmetric about the main diagonal
- Final answer: Two holes diagonally symmetric across the diagonal

Pro Visualization Tip:
Imagine the paper as a book cover. When you open it (unfold), the mark on one side appears mirrored on the other side across the spine (fold line).

Advanced Technique:
For diagonal folds, use coordinate geometry. If the hole is at position (x, y) on the folded triangle, the symmetric hole appears at position (y, x) when unfolded across the diagonal.

Common Pitfall: Don't confuse diagonal symmetry with horizontal/vertical symmetry. Diagonal folds at 45° create patterns along that diagonal axis, not along the edges.

Coordinate Verification:
Original hole: (45,55)
Mirror hole: (55,45) [for top-left to bottom-right diagonal]
Check symmetry: Both equidistant from line y=x

Detailed Step-by-Step Solution:

Step 1 - Diagonal Fold Analysis:
- Fold type: diagonally from top-right to bottom-left
- Fold line: from (100,0) to (0,100) in 100x100 coordinates
- Creates: Triangular shape with top-left corner hidden
- Symmetry axis: Secondary diagonal (top-right to bottom-left)

Step 2 - Coordinate System Setup:
- Origin: top-left corner (0,0)
- Top-right: (100,0)
- Bottom-left: (0,100)
- Bottom-right: (100,100)
- Fold line equation: x + y = 100

Step 3 - Hole Position Mapping:
- Given position: near one corner of the triangle (35,55)
- This is in the visible triangular region (35+55=90 < 100, so below the fold line)
- Distance from fold line: 100-90=10 units away
- Mirror transformation: (x,y) → (100-y, 100-x)

Step 4 - Unfolding Process:
- Fixed layer: maintains hole at (35,55)
- Folded layer: unfolds to reveal mirror hole
- Mirror calculation: (35,55) → (100-55, 100-35) = (45,65)
- Result: Two distinct holes at (35,55) and (45,65)

Step 5 - Pattern Formation:
- Two holes symmetric across the opposite diagonal
- Pattern: Two holes symmetric across the opposite diagonal
- Visual: One in upper-left region, one in lower-right region

Key Learning: Diagonal folds create reflection patterns across the diagonal axis.

Spatial Reasoning: Imagine the diagonal as a mirror - whatever is on one side reflects to the other side.

Detailed Step-by-Step Solution:

Step 1 - Understanding Diagonal Folds:
- Diagonal folds create 45-degree symmetry
- The fold line runs corner to corner (top-left to bottom-right)
- Creates a triangular shape with 2 layers
- Fold direction: diagonally from top-left to bottom-right

Step 2 - Geometric Analysis:
- Original: Square with 4 corners
- After fold: Triangle with 3 visible corners
- Hidden corner: Bottom-right under folded layers
- Symmetry axis: Diagonal line at 45° from top-left to bottom-right
- Layer structure: 2 overlapping triangular layers

Step 3 - Hole Punch Location:
- Position: near the center of folded triangle
- Coordinate mapping: (45,55) in 100x100 coordinate system
- Penetration: Through both triangular layers
- Important: Position relative to diagonal symmetry axis

Step 4 - Mental Unfolding Technique:
- Keep one triangular layer fixed (bottom layer)
- Rotate the other layer 180° around diagonal axis
- The hole on the moving layer traces to its symmetric position
- Use coordinate transformation: (x,y) → (y,x) for this diagonal
- Result: Two holes diagonally symmetric across the diagonal

Step 5 - Pattern Recognition:
- Diagonal symmetry creates diagonal hole pattern
- Both holes equidistant from diagonal fold line
- Perpendicular distance from fold is equal for both holes
- Visual pattern: Two holes symmetric about the main diagonal
- Final answer: Two holes diagonally symmetric across the diagonal

Pro Visualization Tip:
Imagine the paper as a book cover. When you open it (unfold), the mark on one side appears mirrored on the other side across the spine (fold line).

Advanced Technique:
For diagonal folds, use coordinate geometry. If the hole is at position (x, y) on the folded triangle, the symmetric hole appears at position (y, x) when unfolded across the diagonal.

Common Pitfall: Don't confuse diagonal symmetry with horizontal/vertical symmetry. Diagonal folds at 45° create patterns along that diagonal axis, not along the edges.

Coordinate Verification:
Original hole: (45,55)
Mirror hole: (55,45) [for top-left to bottom-right diagonal]
Check symmetry: Both equidistant from line y=x

Detailed Step-by-Step Solution:

Step 1 - Understanding Diagonal Folds:
- Diagonal folds create 45-degree symmetry
- The fold line runs corner to corner (top-left to bottom-right)
- Creates a triangular shape with 2 layers
- Fold direction: diagonally from top-left to bottom-right

Step 2 - Geometric Analysis:
- Original: Square with 4 corners
- After fold: Triangle with 3 visible corners
- Hidden corner: Bottom-right under folded layers
- Symmetry axis: Diagonal line at 45° from top-left to bottom-right
- Layer structure: 2 overlapping triangular layers

Step 3 - Hole Punch Location:
- Position: near the center of folded triangle
- Coordinate mapping: (45,55) in 100x100 coordinate system
- Penetration: Through both triangular layers
- Important: Position relative to diagonal symmetry axis

Step 4 - Mental Unfolding Technique:
- Keep one triangular layer fixed (bottom layer)
- Rotate the other layer 180° around diagonal axis
- The hole on the moving layer traces to its symmetric position
- Use coordinate transformation: (x,y) → (y,x) for this diagonal
- Result: Two holes diagonally symmetric across the diagonal

Step 5 - Pattern Recognition:
- Diagonal symmetry creates diagonal hole pattern
- Both holes equidistant from diagonal fold line
- Perpendicular distance from fold is equal for both holes
- Visual pattern: Two holes symmetric about the main diagonal
- Final answer: Two holes diagonally symmetric across the diagonal

Pro Visualization Tip:
Imagine the paper as a book cover. When you open it (unfold), the mark on one side appears mirrored on the other side across the spine (fold line).

Advanced Technique:
For diagonal folds, use coordinate geometry. If the hole is at position (x, y) on the folded triangle, the symmetric hole appears at position (y, x) when unfolded across the diagonal.

Common Pitfall: Don't confuse diagonal symmetry with horizontal/vertical symmetry. Diagonal folds at 45° create patterns along that diagonal axis, not along the edges.

Coordinate Verification:
Original hole: (45,55)
Mirror hole: (55,45) [for top-left to bottom-right diagonal]
Check symmetry: Both equidistant from line y=x