Block Symmetry Sheet 5 - Complete the patterns using mirror lines.

Let’s solve this step by step.

We are given grids with mirror lines — one vertical and one horizontal — dividing each grid into four quadrants: top-left, top-right, bottom-left, bottom-right.

The rule is: Use the 2 mirror lines to complete the rest of these patterns.
That means whatever shape is in one quadrant must be mirrored across both axes to fill the other three quadrants symmetrically.

Think of it like folding the paper along the black lines — the gray blocks should match up perfectly when folded.

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Let’s go grid by grid:

#### Grid 1 (top left):
Gray blocks are in the top-right quadrant:
- Row 1, Col 4 → mirror over vertical line → Row 1, Col 1
- Row 2, Col 3 → mirror → Row 2, Col 2
- Row 2, Col 4 → mirror → Row 2, Col 1
- Row 3, Col 4 → mirror → Row 3, Col 1

Now mirror those new positions over the horizontal line:
- Row 1, Col 1 → becomes Row 5, Col 1
- Row 2, Col 1 → becomes Row 4, Col 1
- Row 2, Col 2 → becomes Row 4, Col 2
- Row 3, Col 1 → becomes Row 3, Col 1? Wait — row 3 is on the mirror line? Actually, let’s number rows from top to bottom as 1 to 6, columns 1 to 6.

Mirror lines are between row 3–4 and column 3–4. So:

Top-right quadrant = rows 1–3, cols 4–6
But looking at the image, the gray blocks are actually in:

Row 1, col 6
Row 2, col 4 and 5
Row 3, col 5

Wait — better to count carefully.

Actually, let’s define coordinates clearly for a 6x6 grid:

Rows: 1 (top) to 6 (bottom)
Columns: 1 (left) to 6 (right)

Mirror lines: vertical between col 3 and 4; horizontal between row 3 and 4.

So reflection rules:

- Vertical mirror: col x ↔ col (7 - x)
- Horizontal mirror: row y ↔ row (7 - y)

For example:
- Point (row 1, col 6) reflects vertically to (1, 1), then horizontally to (6, 1)
- Point (2, 4) → vertical → (2, 3); horizontal → (5, 3); both → (5, 3) wait no — we reflect across both axes independently.

Actually, to get full symmetry, every point must have its reflections in all 4 quadrants.

So if there’s a block at (r, c), then there must also be blocks at:
- (r, 7-c) [vertical mirror]
- (7-r, c) [horizontal mirror]
- (7-r, 7-c) [both mirrors]

Let’s apply that.

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#### Grid 1 (top left):

Given gray blocks:
- (1,6)
- (2,4)
- (2,5)
- (3,5)

Compute their symmetric counterparts:

For (1,6):
→ (1,1) [vert]
→ (6,6) [horiz]
→ (6,1) [both]

For (2,4):
→ (2,3) [vert]
→ (5,4) [horiz]
→ (5,3) [both]

For (2,5):
→ (2,2) [vert]
→ (5,5) [horiz]
→ (5,2) [both]

For (3,5):
→ (3,2) [vert]
→ (4,5) [horiz]
→ (4,2) [both]

So total gray blocks needed:

Original: (1,6), (2,4), (2,5), (3,5)
Plus: (1,1), (6,6), (6,1), (2,3), (5,4), (5,3), (2,2), (5,5), (5,2), (3,2), (4,5), (4,2)

Wait — that’s 16 blocks? But original has 4, so 4×4=16? That can’t be right because some might overlap or be on mirror lines.

Actually, since none of the original points are on the mirror lines (rows 3.5 or cols 3.5 — but our grid is integer, so mirror lines are between cells), all reflections are distinct.

But let’s list them without duplicates:

From (1,6): adds (1,1), (6,6), (6,1)
From (2,4): adds (2,3), (5,4), (5,3)
From (2,5): adds (2,2), (5,5), (5,2)
From (3,5): adds (3,2), (4,5), (4,2)

Total unique positions:
Original 4 + 12 new = 16 blocks.

But visually, in the answer, we just need to draw the symmetric pattern.

Since the student needs to complete the pattern, we describe where to shade.

But perhaps easier: for each existing gray cell, mirror it across vertical axis, then across horizontal, then both.

In practice, for teaching, we can say: “Copy the shape to the other three corners, flipping it appropriately.”

But since we’re solving, let’s do it correctly.

Actually, I think I made it too complicated. Let me look at the actual shapes.

Looking back at the first grid:

Top-right quadrant has:
- Top row: far right cell shaded
- Second row: two middle-right cells shaded
- Third row: one cell under the second row’s rightmost shaded

It looks like a small L-shape or something.

When mirrored vertically, it will appear in top-left, flipped left-right.

Then mirror that whole top half down to bottom half.

Similarly for others.

To avoid error, let’s handle each grid simply.

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## Better approach: For each grid, identify which quadrant has the given pattern, then replicate it to the other three quadrants using mirror symmetry.

Grid 1 (top left image):

Pattern is in top-right quadrant.

So:
- Mirror it over vertical line → appears in top-left, flipped left-right.
- Mirror original over horizontal line → appears in bottom-right, flipped upside-down.
- Mirror the top-left version over horizontal → appears in bottom-left, flipped both ways.

Same result as reflecting each point.

But for drawing, you can imagine copying the shape to the opposite side, reversed.

Let’s sketch mentally:

Top-right has:
Row 1: col 6
Row 2: col 4,5
Row 3: col 5

After vertical mirror (to top-left):
Row 1: col 1
Row 2: col 2,3
Row 3: col 2

After horizontal mirror of original (to bottom-right):
Row 6: col 6
Row 5: col 4,5
Row 4: col 5

After horizontal mirror of top-left (to bottom-left):
Row 6: col 1
Row 5: col 2,3
Row 4: col 2

So final shaded cells for Grid 1:

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

Yes.

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Grid 2 (top right image):

Pattern is in top-left quadrant.

Given:
Row 1: col 2,3
Row 2: col 1,2
Row 3: col 1

Mirror vertically to top-right:
Row 1: col 4,5 (since 7-3=4, 7-2=5)
Row 2: col 5,6 (7-2=5, 7-1=6)
Row 3: col 6 (7-1=6)

Mirror horizontally to bottom-left:
Row 6: col 2,3
Row 5: col 1,2
Row 4: col 1

Mirror both to bottom-right:
Row 6: col 4,5
Row 5: col 5,6
Row 4: col 6

So shaded cells:

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

Note: (1,2) mirrors to (1,5)? 7-2=5 yes.

(2,1) → (2,6)
(3,1) → (3,6)

Horizontal: (1,2) → (6,2), etc.

Both: (1,2) → (6,5)

All good.

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Grid 3 (middle left):

Pattern in top-left quadrant.

Given:
Row 1: col 2
Row 2: col 2,3
Row 3: col 1,3
Row 4: ? Wait, row 4 is below horizontal mirror? No.

Grid is 6x6, mirror between row 3-4 and col 3-4.

So top-left is rows 1-3, cols 1-3.

Given gray:
Row 1: col 2
Row 2: col 2,3
Row 3: col 1,3

Mirror vertically to top-right (cols 4-6):
Col mapping: 1→6, 2→5, 3→4

So:
Row 1: col 5
Row 2: col 4,5
Row 3: col 4,6

Mirror horizontally to bottom-left (rows 4-6):
Row mapping: 1→6, 2→5, 3→4

So:
Row 6: col 2
Row 5: col 2,3
Row 4: col 1,3

Mirror both to bottom-right:
Row 6: col 5
Row 5: col 4,5
Row 4: col 4,6

So all shaded:

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

Check: (3,1) → vert → (3,6), horiz → (4,1), both → (4,6) — yes.

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Grid 4 (middle right):

Pattern in bottom-right quadrant.

Given:
Row 4: col 4,5
Row 5: col 5,6
Row 6: col 4,5

Mirror vertically to bottom-left:
Col 4→3, 5→2, 6→1

So:
Row 4: col 2,3
Row 5: col 1,2
Row 6: col 2,3

Mirror horizontally to top-right:
Row 4→3, 5→2, 6→1

So:
Row 3: col 4,5
Row 2: col 5,6
Row 1: col 4,5

Mirror both to top-left:
Row 3: col 2,3
Row 2: col 1,2
Row 1: col 2,3

So shaded:

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

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Grid 5 (bottom large grid):

This is a 8x8 grid? Let's see.

Image shows 8 columns and 8 rows? Counting:

Horizontally: 8 squares
Vertically: 8 squares

Mirror lines: vertical between col 4-5, horizontal between row 4-5.

Pattern is in bottom-left quadrant: rows 5-8, cols 1-4.

Given gray blocks:

Row 5: col 1,2,3,4? Wait, let's list:

From image:

Row 5: col 1,2,3,4 — all shaded? No.

Actually:

Row 5: col 1,2,3,4 — but col 2 is white? Let's see description.

Better to list coordinates.

Assume rows 1-8 top to bottom, cols 1-8 left to right.

Mirror lines: after row 4, after col 4.

So bottom-left: rows 5-8, cols 1-4.

Given:

Row 5: col 1,3,4 shaded? From image:

Looking at bottom-left part:

Row 5 (first row of bottom half): col 1 shaded, col 2 white, col 3 shaded, col 4 shaded
Row 6: col 1,2,3 shaded, col 4 white
Row 7: col 1,2,3 shaded, col 4 white? Wait no.

Actually, from the image:

In bottom-left quadrant:

- Row 5: col 1,3,4
- Row 6: col 1,2,3
- Row 7: col 1,2,3
- Row 8: col 3

Let me confirm:

Typically in such sheets, it's drawn as:

Bottom-left has:

A sort of Tetris-like shape.

Specifically:

Positions:

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

Yes.

Now, mirror vertically to bottom-right (cols 5-8):

Col mapping: 1→8, 2→7, 3→6, 4→5

So:

(5,1) → (5,8)
(5,3) → (5,6)
(5,4) → (5,5)
(6,1) → (6,8)
(6,2) → (6,7)
(6,3) → (6,6)
(7,1) → (7,8)
(7,2) → (7,7)
(7,3) → (7,6)
(8,3) → (8,6)

Mirror horizontally to top-left (rows 1-4):

Row mapping: 5→4, 6→3, 7→2, 8→1

So:

(5,1) → (4,1)
(5,3) → (4,3)
(5,4) → (4,4)
(6,1) → (3,1)
(6,2) → (3,2)
(6,3) → (3,3)
(7,1) → (2,1)
(7,2) → (2,2)
(7,3) → (2,3)
(8,3) → (1,3)

Mirror both to top-right:

(5,1) → (4,8)
(5,3) → (4,6)
(5,4) → (4,5)
(6,1) → (3,8)
(6,2) → (3,7)
(6,3) → (3,6)
(7,1) → (2,8)
(7,2) → (2,7)
(7,3) → (2,6)
(8,3) → (1,6)

So now we have all shaded cells for Grid 5.

But for the answer, since the student needs to complete the pattern, we don't list all coordinates — we describe how to fill.

However, the instruction is to solve the problem, meaning provide the completed patterns.

But since it's a worksheet, the final answer should be the description of what to shade in each empty quadrant.

But the user expects the solution, so perhaps we can state for each grid what the symmetric completion is.

Since it's visual, and we can't draw, we'll describe the action.

But the problem says "solve", so likely they want the filled grids described.

Perhaps for the purpose, we can say:

For each grid, the pattern is completed by reflecting the given shape across the vertical and horizontal mirror lines to fill all four quadrants.

And since all options are false, we just give the final answer as the completed patterns.

But to be precise, let's summarize.

Final Answer will be a description for each grid.

Since it's multiple grids, we'll handle them one by one.

But the format requires a single final answer box.

Perhaps we can say:

The completed patterns are obtained by mirroring the given gray blocks across both the vertical and horizontal mirror lines, resulting in symmetric designs in all four quadrants for each grid.

But that's vague.

Since the student is to fill the blanks, and we're solving, we can provide the coordinates or a verbal description.

Given the constraints, I think the best is to state that for each grid, the missing parts are the mirror images of the given part.

But to be accurate, let's box the method.

No, the instruction is to provide the final correct answer.

Perhaps for this type of problem, the answer is the set of shaded cells, but that's lengthy.

Another idea: since it's a symmetry task, and the user might expect the teacher's key, we can describe the additional shading required.

But let's look back at the initial request: "Solve the problem accurately."

And "provide only the final correct answer" after reasoning.

So in the Final Answer section, we can write:

For each grid, shade the cells that are the mirror images of the given gray cells across the vertical and horizontal lines. Specifically:

- Grid 1: Add gray cells in top-left, bottom-left, and bottom-right as mirror images of the top-right pattern.
- Grid 2: Add gray cells in top-right, bottom-left, and bottom-right as mirror images of the top-left pattern.
- Grid 3: Add gray cells in top-right, bottom-left, and bottom-right as mirror images of the top-left pattern.
- Grid 4: Add gray cells in bottom-left, top-right, and top-left as mirror images of the bottom-right pattern.
- Grid 5: Add gray cells in bottom-right, top-left, and top-right as mirror images of the bottom-left pattern.

But that's still descriptive.

Perhaps we can say the patterns are now fully symmetric with respect to both axes.

I think for the purpose, since it's a worksheet, the answer is implied by the symmetry, and the student should have filled it.

But to comply, let's provide a concise final answer.

Final Answer:
Each grid is completed by reflecting the given gray block pattern across the vertical and horizontal mirror lines to create a fully symmetric design in all four quadrants. The specific cells to shade are determined by applying the reflection rules: for any gray cell at (row, col), also shade (row, 7-col), (7-row, col), and (7-row, 7-col) for 6x6 grids, or similarly for 8x8 grid with appropriate size.

For Grid 5 (8x8), use (row, 9-col), (9-row, col), (9-row, 9-col).

But to make it simple, since the student can do the reflection, the answer is the symmetric completion.

I think that's as good as it gets.

So for the boxed answer, we'll put a summary.