Key takeaway: A reflection produces a mirror image of a shape across a chosen line, with every point landing as far behind the line as it was in front.
Grade 88.G.A.1Geometry and Measurement
Definition of Reflection
A reflection acts like a mirror. Every point of the figure jumps to the opposite side of a chosen line, landing exactly as far from the line as it started. The segment joining a point to its image crosses the line at a right angle and is cut in half by it.
The size and shape stay the same, so the image is congruent to the original, but the orientation reverses: if you read the vertices clockwise in the original, you read them counterclockwise in the image. Points that lie on the line of reflection do not move. On a coordinate grid the two most common mirrors are the x-axis and the y-axis, and each has a simple sign rule.
Formula and notation
For the two axes, the rules are
\[\text{over the } y\text{-axis: } (x, y) \to (-x, y) \qquad \text{over the } x\text{-axis: } (x, y) \to (x, -y)\]
The image of \(P\) is written \(P^{\prime}\).
Key facts
- Distances from the mirror line match on both sides of it.
- Lengths and angles survive the flip, but left-right order reverses.
- A point sitting on the mirror line does not move at all.
- Using the vertical axis as mirror changes only the sign of x.
Reflection example
Triangle \(PQR\) has corners \(P(2, 1)\), \(Q(6, 1)\) and \(R(4, 5)\). Flip it across the vertical axis and list the new corners. Then compare how far \(R\) sits from the mirror before and after.
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Use the rule \((x, y) \to (-x, y)\).
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\(P(2, 1) \to P^{\prime}(-2, 1)\), \(Q(6, 1) \to Q^{\prime}(-6, 1)\), \(R(4, 5) \to R^{\prime}(-4, 5)\).
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R is 4 units from the y-axis, and \(R^{\prime}\) is also 4 units from it, on the left.
The image is \(P^{\prime}(-2, 1)\), \(Q^{\prime}(-6, 1)\), \(R^{\prime}(-4, 5)\).
Common mistake
Watch out
Reflecting \(R(4, 5)\) over the y-axis, Dev writes \(R^{\prime}(4, -5)\).
Why it is wrong
That is the rule for the x-axis. A mirror along the vertical y-axis flips left and right, so only x changes sign: \(R^{\prime}(-4, 5)\). Picture which way the point has to jump: across the vertical line, not up or down.
Related terms and lessons
- Grade 8 · Translations, Reflections and RotationsLessonWorksheetQuiz
- Geometry · Rigid TransformationsLessonWorksheetQuiz
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