Symmetry Worksheet - Free Printable

Let's solve the Block Symmetry Sheet 8 step by step.

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🔍 Objective:

Use the diagonal lines of symmetry to complete the missing patterns in each grid. The diagonal line acts as a mirror — whatever is on one side must be mirrored across the diagonal to complete the pattern.

There are 6 grids, each with a diagonal line (either from top-left to bottom-right or top-right to bottom-left). We need to reflect the shaded blocks across this diagonal to complete the symmetric shape.

We’ll go through each grid one by one.

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## ✔ Grid 1 (Top Left)

- Diagonal: Top-left to bottom-right (↘)
- Shaded blocks are on the left side of the diagonal.
- We need to reflect them across the diagonal to the right side.

Step-by-step:

Look at each shaded square and find its mirror image across the diagonal.

For example:
- A shaded square at (row 2, col 1) → reflected to (row 1, col 2)
- (row 3, col 2) → (row 2, col 3)
- (row 4, col 3) → (row 3, col 4)
- (row 5, col 1) → (row 1, col 5)

Wait — let’s label rows from top to bottom (1 to 6), columns left to right (1 to 6).

Current shaded squares:
- (2,1)
- (3,2)
- (4,3)
- (5,1)

Now reflect across the main diagonal (where row = col):

Reflection rule:
If a block is at (r,c), its reflection across the diagonal is at (c,r).

So:
- (2,1) → (1,2)
- (3,2) → (2,3)
- (4,3) → (3,4)
- (5,1) → (1,5)

Now shade these:
- (1,2), (2,3), (3,4), (1,5)

✔ So add these on the right side.

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## ✔ Grid 2 (Top Right)

- Diagonal: Top-right to bottom-left (↙)
- This is the anti-diagonal.
- Reflection rule: For a point (r,c), its mirror across anti-diagonal is (7−c, 7−r) in a 6×6 grid.

But easier: just count distance from the diagonal.

Alternatively, think: if you have a block at (r,c), then its reflection across the anti-diagonal is at (7−c, 7−r).

Let’s list shaded blocks:

Shaded:
- (2,5)
- (3,5)
- (3,6)
- (4,4)
- (5,3)
- (5,2)

Now reflect each across the anti-diagonal:

Use formula: (r,c) → (7−c, 7−r)

- (2,5) → (7−5, 7−2) = (2,5) → already on diagonal? Wait no — (2,5): r=2, c=5 → (7−5, 7−2) = (2,5)? No: (2,5) → (2,5)? That can't be.

Wait: anti-diagonal is where r + c = 7.

So reflection across anti-diagonal: (r,c) → (7−c, 7−r)

Check:
- (2,5): r=2, c=5 → (7−5, 7−2) = (2,5) → same point? But (2+5)=7 → it's on the diagonal! So it stays.

Wait — actually (2,5): 2+5=7 → yes, it's on the anti-diagonal → so it's fixed.

But wait: (2,5) is on the diagonal → so we don’t need to reflect it — it's already there.

But let’s check others:

- (3,5): r=3, c=5 → (7−5, 7−3) = (2,4)
- (3,6): (7−6, 7−3) = (1,4)
- (4,4): (7−4, 7−4) = (3,3)
- (5,3): (7−3, 7−5) = (4,2)
- (5,2): (7−2, 7−5) = (5,2) → (5+2)=7 → also on diagonal → fixed

So reflections:
- (3,5) → (2,4)
- (3,6) → (1,4)
- (4,4) → (3,3)
- (5,3) → (4,2)

Now add:
- (2,4), (1,4), (3,3), (4,2)

These should be shaded.

Note: (2,5), (5,2) are already on the diagonal → stay.

✔ So fill in (1,4), (2,4), (3,3), (4,2)

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## ✔ Grid 3 (Middle Left)

- Diagonal: Top-left to bottom-right (↘)
- Shaded blocks on the right side of diagonal
- Need to reflect across diagonal to the left side

Shaded blocks:
- (2,3), (2,4), (2,5), (2,6)
- (3,4), (3,5), (3,6)
- (4,5), (4,6)
- (5,6)

Now reflect each across diagonal: (r,c) → (c,r)

So:
- (2,3) → (3,2)
- (2,4) → (4,2)
- (2,5) → (5,2)
- (2,6) → (6,2)
- (3,4) → (4,3)
- (3,5) → (5,3)
- (3,6) → (6,3)
- (4,5) → (5,4)
- (4,6) → (6,4)
- (5,6) → (6,5)

Add all these on the left side:
- (3,2), (4,2), (5,2), (6,2)
- (4,3), (5,3), (6,3)
- (5,4), (6,4)
- (6,5)

✔ Shade these.

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## ✔ Grid 4 (Middle Right)

- Diagonal: Top-right to bottom-left (↙) → anti-diagonal
- Shaded blocks on the bottom-left side
- Reflect across anti-diagonal to top-right

Shaded:
- (4,1), (5,1), (6,1)
- (4,2), (5,2), (6,2)
- (4,3), (5,3), (6,3)
- (5,4), (6,4)
- (6,5)

Now reflect using: (r,c) → (7−c, 7−r)

Compute:
- (4,1): (7−1, 7−4) = (6,3)
- (5,1): (7−1, 7−5) = (6,2)
- (6,1): (7−1, 7−6) = (6,1) → on diagonal → stays
- (4,2): (7−2, 7−4) = (5,3)
- (5,2): (7−2, 7−5) = (5,2) → on diagonal → stays
- (6,2): (7−2, 7−6) = (5,1)
- (4,3): (7−3, 7−4) = (4,3) → on diagonal → stays
- (5,3): (7−3, 7−5) = (4,2)
- (6,3): (7−3, 7−6) = (4,1)
- (5,4): (7−4, 7−5) = (3,2)
- (6,4): (7−4, 7−6) = (3,1)
- (6,5): (7−5, 7−6) = (2,1)

Now list new points:
- (6,3), (6,2), (5,3), (5,2), (5,1), (4,3), (4,2), (4,1), (3,2), (3,1), (2,1)

But many are already shaded or on diagonal.

The ones not already present:
- (3,2), (3,1), (2,1)

So we need to shade:
- (3,2), (3,1), (2,1)

Also, check if others are already there.

Wait — but original has (5,4), (6,4), (6,5) — their reflections:
- (5,4) → (3,2)
- (6,4) → (3,1)
- (6,5) → (2,1)

So yes, add (3,2), (3,1), (2,1)

✔ Add those.

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## ✔ Grid 5 (Bottom Left)

- Diagonal: Top-left to bottom-right (↘)
- Shaded on top-left side
- Reflect to bottom-right

Shaded:
- (1,1), (1,2), (1,3), (1,4)
- (2,2), (2,3), (2,4)
- (3,3), (3,4)
- (4,4)
- (5,5)
- (6,6)

Wait — (5,5), (6,6) are on diagonal → stay

Others:
- (1,1) → (1,1)
- (1,2) → (2,1)
- (1,3) → (3,1)
- (1,4) → (4,1)
- (2,2) → (2,2)
- (2,3) → (3,2)
- (2,4) → (4,2)
- (3,3) → (3,3)
- (3,4) → (4,3)
- (4,4) → (4,4)

So reflections:
- (2,1), (3,1), (4,1), (3,2), (4,2), (4,3)

Add these on the bottom-right side.

✔ Shade:
- (2,1), (3,1), (4,1), (3,2), (4,2), (4,3)

(And keep (5,5), (6,6) as they are on diagonal.)

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## ✔ Grid 6 (Bottom Right)

- Diagonal: Top-right to bottom-left (↙) → anti-diagonal
- Shaded on top-right side
- Reflect to bottom-left

Shaded:
- (1,1), (1,2), (1,3), (1,4), (1,5)
- (2,1), (2,2), (2,3), (2,4)
- (3,1), (3,2), (3,3)
- (4,1), (4,2)
- (5,1)

Now reflect across anti-diagonal: (r,c) → (7−c, 7−r)

Compute:
- (1,1): (6,6)
- (1,2): (5,6)
- (1,3): (4,6)
- (1,4): (3,6)
- (1,5): (2,6)
- (2,1): (6,5)
- (2,2): (5,5)
- (2,3): (4,5)
- (2,4): (3,5)
- (3,1): (6,4)
- (3,2): (5,4)
- (3,3): (4,4)
- (4,1): (6,3)
- (4,2): (5,3)
- (5,1): (6,2)

So the reflections are:
- (6,6), (5,6), (4,6), (3,6), (2,6)
- (6,5), (5,5), (4,5), (3,5)
- (6,4), (5,4), (4,4)
- (6,3), (5,3)
- (6,2)

Now add these on the bottom-left side.

✔ Shade all these.

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## 🧩 Summary of Solutions:

Grid 1:

- Add: (1,2), (2,3), (3,4), (1,5)

Grid 2:

- Add: (1,4), (2,4), (3,3), (4,2)

Grid 3:

- Add: (3,2), (4,2), (5,2), (6,2), (4,3), (5,3), (6,3), (5,4), (6,4), (6,5)

Grid 4:

- Add: (2,1), (3,1), (3,2)

Grid 5:

- Add: (2,1), (3,1), (4,1), (3,2), (4,2), (4,3)

Grid 6:

- Add: (2,6), (3,6), (4,6), (5,6), (6,6), (3,5), (4,5), (5,5), (6,5), (4,4), (5,4), (6,4), (5,3), (6,3), (6,2)

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## ✔ Final Answer:

You should shade the following additional blocks in each grid to complete the symmetry:

| Grid | Add These Coordinates |
|------|------------------------|
| 1 | (1,2), (2,3), (3,4), (1,5) |
| 2 | (1,4), (2,4), (3,3), (4,2) |
| 3 | (3,2), (4,2), (5,2), (6,2), (4,3), (5,3), (6,3), (5,4), (6,4), (6,5) |
| 4 | (2,1), (3,1), (3,2) |
| 5 | (2,1), (3,1), (4,1), (3,2), (4,2), (4,3) |
| 6 | (2,6), (3,6), (4,6), (5,6), (6,6), (3,5), (4,5), (5,5), (6,5), (4,4), (5,4), (6,4), (5,3), (6,3), (6,2) |

> ⚠️ Note: Use a pencil to shade in the grid on the worksheet, ensuring that each shape is symmetric across the given diagonal.

Let me know if you'd like a visual version or printable answer key!