Flipping across a line
A reflection produces a mirror image of a shape across a straight line, the mirror line. Every point on the image sits directly opposite its original, the same distance from the line but on the other side. The shape stays the same size, so the image is congruent, but it is flipped, so a shape and its reflection have opposite handedness.
- Mirror line
- The line a shape is reflected in. Each point and its image are the same perpendicular distance from it, on opposite sides. Points that already lie on the mirror line do not move.
Common mirror lines
Reflecting in the axes and in vertical or horizontal lines follows tidy coordinate rules:
| Mirror line | Effect on \((x,y)\) |
|---|---|
| The \(x\)-axis (the line \(y=0\)) | \((x,y) \to (x,\ -y)\) |
| The \(y\)-axis (the line \(x=0\)) | \((x,y) \to (-x,\ y)\) |
| A vertical line \(x=a\) | the \(x\)-coordinate flips about \(a\) |
| A horizontal line \(y=b\) | the \(y\)-coordinate flips about \(b\) |
Measure at right angles
The distance from a point to the mirror line is always measured perpendicular to the line, not slanted. Count the perpendicular gap, then step the same gap out on the far side.
The diagonal lines
Two diagonal mirrors come up often, and both have simple rules. Reflecting in \(y=x\) swaps the coordinates over. Reflecting in \(y=-x\) swaps them and changes both signs.
Do not confuse the two diagonals
Reflection in \(y=x\) just swaps \(x\) and \(y\). Reflection in \(y=-x\) swaps them and makes both negative. Write the rule down before you start so the signs do not catch you out.
A worked example
Triangle \(ABC\) has vertices \(A(1,2)\), \(B(4,2)\) and \(C(1,5)\). Reflect it in the line \(y=x\). Find the image coordinates.
Where this is assessed
Applying the correct rule is Criterion A (Knowing and understanding). A full description names the transformation as a reflection and states the equation of the mirror line, which is Criterion C (Communicating). Just saying reflection is not enough, the line must be named. Symmetry in design and nature reaches Criterion D (Applying maths in context).
Check yourself
1. Reflect the point (3,-2) in the x-axis. Where does it go? +
Reflecting in the \(x\)-axis uses \((x,y) \to (x,-y)\). So \((3,-2) \to \mathbf{(3,2)}\).
2. Reflect the point (6,1) in the line y = x. Where does it land? +
Reflecting in \(y=x\) swaps the coordinates: \((6,1) \to \mathbf{(1,6)}\).
3. Reflect the point (2,5) in the y-axis. Where does it go? +
Reflecting in the \(y\)-axis uses \((x,y) \to (-x,y)\). So \((2,5) \to \mathbf{(-2,5)}\).
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