P4 Line Symmetry

A line of symmetry divides a figure into two matching halves. Each point and its mirror are the same perpendicular distance from the line, on opposite sides. The line itself is the axis.

Parents: ask your child to test every part of a proposed axis. A shape that looks centred can still have one unmatched corner.

By the end you will be able to identify symmetric figures, test a straight axis and complete a symmetric figure on a square grid.

Identify a matching half

Imagine folding along the proposed line. Every filled part on one side must land exactly on a filled part on the other side. A vertical axis through the middle of an 8 by 8 grid pairs column 3 with column 6, column 1 with column 8 and so on. A horizontal axis pairs row 2 with row 7.

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Symmetry grid lab: reflect across an axis

🤔 Predict first: A filled cell is in column 3, row 2. Across a vertical middle line, where is its mirror?

column 6, row 2column 5, row 2column 3, row 7

Text alternative: the lab uses an 8 by 8 square grid with a visible vertical or horizontal axis. It lists each filled square and shows an open circle at a missing mirror. A diagonal or tilted axis follows the same perpendicular-distance rule; it does not require a new formula.

Pair the cells

$$\text{vertical axis: column 3}\leftrightarrow\text{column 6 in the same row}$$

$$\text{horizontal axis: row 2}\leftrightarrow\text{row 7 in the same column}$$

$$\text{every filled cell needs one matching partner}$$

Test an alleged axis

To test a straight line, choose each filled part in turn and look for its reflected partner. The line is a valid axis only when all partners match, no partner is outside the figure and no unmatched part remains. A visual centre is not enough.

For a diagonal example, pair cells at matching perpendicular distances on opposite sides of the tilted line. For example, one pair can be mirrored across a diagonal as two equally distant cells; the test is still “fold and match”.

Complete a grid figure

Suppose an 8 by 8 grid has a vertical axis and one filled square in column 1, row 7. Its missing mirror is column 8, row 7. Fill that square to complete the pair. If both column 1, row 7 and column 8, row 7 are already filled, that pair is complete.

Practise with worked steps

Practice givens 1: An 8 by 8 grid has a vertical middle axis; the filled cell is column 3, row 2. Find and fill its mirror cell.

Reveal solution: Practice 1

Given a vertical middle axis on an 8 by 8 grid, the filled cell is column 3, row 2. Its mirror is column 6, row 2. The row is unchanged and the perpendicular distances match.

Reflect the cell

$$8+1-3=6$$

$$\text{mirror}=\text{column 6, row 2}$$

Practice givens 2: Across a vertical axis, cells column 1, row 7 and column 8, row 7 are filled. Decide whether this pair needs another cell.

Reveal solution: Practice 2

Given a vertical axis, filled cells are column 1, row 7 and column 8, row 7. The pair is already complete, so no additional cell is needed for that pair.

Check the pair

$$\text{column 1, row 7}\leftrightarrow\text{column 8, row 7}$$

$$\text{both partners are filled}\Rightarrow\text{complete}$$

Practice givens 3: A vertical axis has filled cells column 2, row 5 and column 7, row 5, plus column 4, row 3 with no partner. Test the proposed axis and identify the missing mirror.

Reveal solution: Practice 3

Given filled cells column 2, row 5 and column 7, row 5 across a vertical axis, plus column 4, row 3 with no partner, the line is not a line of symmetry. The missing partner of column 4, row 3 is column 5, row 3.

Find the missing partner

$$8+1-4=5$$

$$\text{column 4, row 3}\leftrightarrow\text{column 5, row 3}$$

$$\text{unmatched cell}\Rightarrow\text{axis is invalid}$$

Watch out

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. What must happen in a line-symmetric figure?

   One side may have an unmatched cornerEach part has a matching mirror across the axisThe axis can be curved

2. Across a vertical middle line, column 2 pairs with which column?

   Column 6Column 2Column 7

3. A filled cell is column 1, row 7. What is its vertical mirror?

   Column 7, row 8Column 1, row 2Column 8, row 7

4. An alleged axis leaves one filled cell without a partner. What is the verdict?

   It is always validIt is not a line of symmetryOnly the grid size decides

5. A horizontal axis pairs row 2 with which row on an 8-row grid?

   Row 6Row 7Row 2

6. What rule also works for a diagonal or tilted axis?

   Count only the top rowEqual perpendicular distances on opposite sidesUse the longest side

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