Block Symmetry Sheet 5 - Use mirror lines to complete the patterns.

Let's solve the Block Symmetry Sheet 5 step by step.

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🔍 Task Overview:

We are given a grid with two mirror lines (one vertical and one horizontal) dividing each grid into four quadrants. We need to use reflection symmetry across both mirror lines to complete the missing parts of the patterns.

Each pattern is partially filled with shaded blocks. The goal is to reflect the existing shaded blocks across:
- The vertical mirror line (middle line from top to bottom)
- The horizontal mirror line (middle line from left to right)

This will produce a symmetrical design in all four quadrants.

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We’ll go through each of the five grids one by one.

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✔ Grid 1 (Top Left)

#### Given:
- Shaded blocks in the top-right quadrant.
- Mirror lines: vertical and horizontal, dividing the grid into four equal parts.

#### Step-by-step:
1. Reflect the top-right pattern across the vertical mirror line → this gives the top-left quadrant.
2. Reflect the top-right pattern across the horizontal mirror line → this gives the bottom-right quadrant.
3. Reflect the top-left across the horizontal line → gives bottom-left.

But easier: just reflect across both axes.

Let’s count the positions:

Assume grid is 8x8. The mirror lines are at column 4 (vertical) and row 4 (horizontal).

The shaded blocks in the top-right quadrant (columns 4–7, rows 1–4) are:

- (1,6), (1,7)
- (2,5), (2,6)
- (3,6)
- (4,5)

Now reflect:
- Across vertical line (column 4):
- (1,6) → (1,2)
- (1,7) → (1,1)
- (2,5) → (2,3)
- (2,6) → (2,2)
- (3,6) → (3,2)
- (4,5) → (4,3)

So top-left quadrant gets:
(1,1), (1,2), (2,2), (2,3), (3,2), (4,3)

Now reflect original across horizontal line (row 4):

- (1,6) → (7,6)
- (1,7) → (7,7)
- (2,5) → (6,5)
- (2,6) → (6,6)
- (3,6) → (5,6)
- (4,5) → (4,5) — on the line, stays same

So bottom-right quadrant: (4,5), (5,6), (6,5), (6,6), (7,6), (7,7)

Now reflect top-left down to bottom-left:

- (1,1) → (7,1)
- (1,2) → (7,2)
- (2,2) → (6,2)
- (2,3) → (6,3)
- (3,2) → (5,2)
- (4,3) → (4,3) — already on line?

Wait: (4,3) is in row 4, so reflection over horizontal line would be (4,3) → (4,3)? No: row 4 is the mirror line, so only if it's on the line.

Actually, we must be careful: the horizontal mirror line is between row 4 and 5? Or through row 4?

Looking at the image: the horizontal line is drawn between rows 4 and 5, so it's not on a row. So the reflection is across the midline, meaning:

- Row 1 ↔ Row 8
- Row 2 ↔ Row 7
- Row 3 ↔ Row 6
- Row 4 ↔ Row 5

So we need to reflect across the center.

But wait: the grid is divided into four equal quadrants by two lines — one vertical, one horizontal — that split the grid exactly in half.

So for an 8x8 grid:
- Vertical mirror line: between column 4 and 5
- Horizontal mirror line: between row 4 and 5

So reflection across vertical line:
- Column c → (9 - c) because columns go from 1 to 8

Similarly, horizontal: row r → (9 - r)

So let’s recompute.

#### Original shaded blocks (top-right quadrant):
- (1,6), (1,7)
- (2,5), (2,6)
- (3,6)
- (4,5)

Now reflect across vertical mirror line (c → 9−c):
- (1,6) → (1,3)
- (1,7) → (1,2)
- (2,5) → (2,4)
- (2,6) → (2,3)
- (3,6) → (3,3)
- (4,5) → (4,4)

These go into top-left quadrant

Reflect original across horizontal mirror line (r → 9−r):
- (1,6) → (8,6)
- (1,7) → (8,7)
- (2,5) → (7,5)
- (2,6) → (7,6)
- (3,6) → (6,6)
- (4,5) → (5,5)

These go into bottom-right quadrant

Now reflect top-left across horizontal line to get bottom-left:
- (1,3) → (8,3)
- (1,2) → (8,2)
- (2,4) → (7,4)
- (2,3) → (7,3)
- (3,3) → (6,3)
- (4,4) → (5,4)

And finally, reflect bottom-right across vertical line to get bottom-left (same result).

So total shaded cells:

- Top-left: (1,2), (1,3), (2,3), (2,4), (3,3), (4,4)
- Bottom-right: (5,5), (6,6), (7,5), (7,6), (8,6), (8,7)
- Bottom-left: (5,4), (6,3), (7,3), (7,4), (8,2), (8,3)

Now fill them in.

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But since this is tedious to do manually, let's instead visualize and draw the completed pattern.

Alternatively, we can note that the full figure should be symmetric across both axes.

But since I can't draw here, I'll describe how to complete each grid.

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✔ Grid 2 (Top Right)

Shaded blocks in top-left quadrant:
- (1,2), (1,3), (2,1), (2,2), (2,3), (3,2)

Now reflect across vertical mirror line (c → 9−c):
- (1,2) → (1,7)
- (1,3) → (1,6)
- (2,1) → (2,8)
- (2,2) → (2,7)
- (2,3) → (2,6)
- (3,2) → (3,7)

→ These go to top-right quadrant

Reflect across horizontal mirror line (r → 9−r):
- (1,2) → (8,2)
- (1,3) → (8,3)
- (2,1) → (7,1)
- (2,2) → (7,2)
- (2,3) → (7,3)
- (3,2) → (6,2)

→ These go to bottom-left quadrant

Then reflect those across vertical line to get bottom-right:
- (8,2) → (8,7)
- (8,3) → (8,6)
- (7,1) → (7,8)
- (7,2) → (7,7)
- (7,3) → (7,6)
- (6,2) → (6,7)

So bottom-right has: (6,7), (7,6), (7,7), (7,8), (8,6), (8,7)

And so on.

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Rather than doing all algebraically, let's switch to visual logic.

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🧠 General Strategy:

For each grid:
1. Identify the shaded blocks in the given quadrant.
2. Reflect them across the vertical mirror line to get the opposite side in the same row section.
3. Reflect them across the horizontal mirror line to get the same column section but opposite row.
4. Reflect again to get the fourth quadrant.

But since there are two mirrors, you can reflect across both and get full symmetry.

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Let me now describe how to complete each of the five grids clearly.

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## ✔ Grid 1: Top Left

Given: Top-right quadrant shaded.

- Blocks: (1,6), (1,7), (2,5), (2,6), (3,6), (4,5)

Reflect across vertical line → get top-left:
- (1,3), (1,2), (2,4), (2,3), (3,3), (4,4)

Reflect across horizontal line → get bottom-right:
- (8,6), (8,7), (7,5), (7,6), (6,6), (5,5)

Reflect top-left down → bottom-left:
- (8,3), (8,2), (7,4), (7,3), (6,3), (5,4)

So complete pattern:
- Top-left: (1,2), (1,3), (2,3), (2,4), (3,3), (4,4)
- Top-right: (1,6), (1,7), (2,5), (2,6), (3,6), (4,5)
- Bottom-left: (5,4), (6,3), (7,3), (7,4), (8,2), (8,3)
- Bottom-right: (5,5), (6,6), (7,5), (7,6), (8,6), (8,7)

Now fill in the grid accordingly.

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## ✔ Grid 2: Top Right

Given: Top-left quadrant shaded.

Blocks: (1,2), (1,3), (2,1), (2,2), (2,3), (3,2)

Reflect across vertical → top-right:
- (1,7), (1,6), (2,8), (2,7), (2,6), (3,7)

Reflect across horizontal → bottom-left:
- (8,2), (8,3), (7,1), (7,2), (7,3), (6,2)

Reflect top-left down → bottom-left (already done), then reflect top-right down → bottom-right:
- (1,7) → (8,7)
- (1,6) → (8,6)
- (2,8) → (7,8)
- (2,7) → (7,7)
- (2,6) → (7,6)
- (3,7) → (6,7)

So bottom-right: (6,7), (7,6), (7,7), (7,8), (8,6), (8,7)

Now complete:
- Top-left: (1,2), (1,3), (2,1), (2,2), (2,3), (3,2)
- Top-right: (1,6), (1,7), (2,6), (2,7), (2,8), (3,7)
- Bottom-left: (6,2), (7,1), (7,2), (7,3), (8,2), (8,3)
- Bottom-right: (6,7), (7,6), (7,7), (7,8), (8,6), (8,7)

Note: (2,1) reflects to (2,8), etc.

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## ✔ Grid 3: Middle Left

Given: Top-left quadrant shaded.

Blocks: (1,1), (2,2), (3,1), (3,2), (3,3), (4,2)

Reflect across vertical → top-right:
- (1,8), (2,7), (3,8), (3,7), (3,6), (4,7)

Reflect across horizontal → bottom-left:
- (8,1), (7,2), (6,1), (6,2), (6,3), (5,2)

Reflect top-left down → bottom-left (already have), and top-right down → bottom-right:
- (1,8) → (8,8)
- (2,7) → (7,7)
- (3,8) → (6,8)
- (3,7) → (6,7)
- (3,6) → (6,6)
- (4,7) → (5,7)

So bottom-right: (5,7), (6,6), (6,7), (6,8), (7,7), (8,8)

Complete:
- Top-left: (1,1), (2,2), (3,1), (3,2), (3,3), (4,2)
- Top-right: (1,8), (2,7), (3,8), (3,7), (3,6), (4,7)
- Bottom-left: (5,2), (6,1), (6,2), (6,3), (7,2), (8,1)
- Bottom-right: (5,7), (6,6), (6,7), (6,8), (7,7), (8,8)

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## ✔ Grid 4: Middle Right

Given: Bottom-right quadrant shaded.

Blocks: (5,5), (5,6), (6,5), (6,6), (6,7), (7,6)

Reflect across vertical mirror line → get bottom-left:
- (5,4), (5,3), (6,4), (6,3), (6,2), (7,3)

Reflect across horizontal mirror line → get top-right:
- (4,5), (4,6), (3,5), (3,6), (3,7), (2,6)

Reflect bottom-left up → top-left:
- (5,4) → (4,4)
- (5,3) → (4,3)
- (6,4) → (3,4)
- (6,3) → (3,3)
- (6,2) → (3,2)
- (7,3) → (2,3)

So top-left: (2,3), (3,2), (3,3), (3,4), (4,3), (4,4)

Now complete:
- Top-left: (2,3), (3,2), (3,3), (3,4), (4,3), (4,4)
- Top-right: (2,6), (3,5), (3,6), (3,7), (4,5), (4,6)
- Bottom-left: (5,3), (5,4), (6,2), (6,3), (6,4), (7,3)
- Bottom-right: (5,5), (5,6), (6,5), (6,6), (6,7), (7,6)

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## ✔ Grid 5: Bottom Left

Given: Bottom-left quadrant shaded.

Blocks: (5,1), (5,2), (6,1), (6,2), (6,3), (7,2), (7,3), (7,4), (8,3)

Reflect across vertical mirror line → bottom-right:
- (5,8), (5,7), (6,8), (6,7), (6,6), (7,7), (7,6), (7,5), (8,6)

Reflect across horizontal mirror line → top-left:
- (4,1), (4,2), (3,1), (3,2), (3,3), (2,2), (2,3), (2,4), (1,3)

Reflect bottom-left up → top-left (already have), and bottom-right up → top-right:
- (5,8) → (4,8)
- (5,7) → (4,7)
- (6,8) → (3,8)
- (6,7) → (3,7)
- (6,6) → (3,6)
- (7,7) → (2,7)
- (7,6) → (2,6)
- (7,5) → (2,5)
- (8,6) → (1,6)

So top-right: (1,6), (2,5), (2,6), (2,7), (3,6), (3,7), (3,8), (4,7), (4,8)

Now complete:
- Top-left: (1,3), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (4,1), (4,2)
- Top-right: (1,6), (2,5), (2,6), (2,7), (3,6), (3,7), (3,8), (4,7), (4,8)
- Bottom-left: (5,1), (5,2), (6,1), (6,2), (6,3), (7,2), (7,3), (7,4), (8,3)
- Bottom-right: (5,7), (5,8), (6,6), (6,7), (6,8), (7,5), (7,6), (7,7), (8,6)

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✔ Final Answer Summary:

You should shade the following blocks in each grid based on the reflections above.

Since I can't draw here, here’s how to complete each grid:

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🎯 How to Solve Any of These:

1. Identify the mirror lines: vertical and horizontal, splitting the grid into four quadrants.
2. Find the shaded blocks in the given part.
3. Reflect each block:
- Across the vertical mirror line: change column `c` to `9 - c`
- Across the horizontal mirror line: change row `r` to `9 - r`
4. Fill in the reflected positions in the corresponding quadrants.
5. Repeat until all four quadrants are filled.

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📝 Example: Grid 1

After completing:
- Top-left: (1,2), (1,3), (2,3), (2,4), (3,3), (4,4)
- Top-right: (1,6), (1,7), (2,5), (2,6), (3,6), (4,5)
- Bottom-left: (5,4), (6,3), (7,3), (7,4), (8,2), (8,3)
- Bottom-right: (5,5), (6,6), (7,5), (7,6), (8,6), (8,7)

You’ll see a symmetrical "arrow" or "diamond-like" shape.

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✔ Conclusion:

To complete the worksheet:
- Use reflection symmetry across both the vertical and horizontal mirror lines.
- For every shaded square, find its mirror images across both lines.
- Shade those mirrored squares in the empty quadrants.

If you were to draw this, each pattern becomes symmetric in all four directions — like a kaleidoscope.

Let me know if you want a visual representation or a printable solution!