Key takeaway: A rigid transformation is a move of a figure in the plane that preserves every distance and every angle, so the image is congruent to the original; translations, reflections and rotations all qualify.
Grade 10 – GeometryG-CO.A.5Geometry and Measurement
Definition of Rigid Transformation
A rigid transformation slides, flips or turns a figure without stretching, shrinking or bending it. Each point moves to a new location, but the distance between any two points stays the same, and so does every angle.
That is why the picture after the move is congruent to the figure before it. The three basic kinds are translations (slides), reflections (flips) and rotations (turns), and any chain of them is also rigid. A dilation is not rigid unless its scale factor is 1, since it changes size.
Formula and notation
A transformation maps each point \(P\) to its image \(P^{\prime}\). It is rigid when, for every pair of points:
\[ P^{\prime}Q^{\prime} = PQ \]
A translation by \(a\) right and \(b\) up is written \((x, y) \to (x + a,\ y + b)\). The reflection across the \(y\)-axis is \((x, y) \to (-x,\ y)\).
Key facts
- Side lengths, angle measures, perimeter and area all stay the same.
- Reflections reverse orientation; translations and rotations keep it.
- A figure and its image under a rigid motion are congruent.
- Dilations with scale factor other than 1 change size, so they are not rigid.
Rigid Transformation example
Segment \(AB\) has \(A(2, 1)\) and \(B(7, 3)\). Rule \(R\) sends \((x, y)\) to \((x + 4,\ y – 3)\). Rule \(S\) sends \((x, y)\) to \((2x,\ y)\). Which rule is a rigid transformation?
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Original length: \(AB = \sqrt{(7-2)^2 + (3-1)^2} = \sqrt{29}\).
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Under \(R\): \(A^{\prime}(6, -2)\), \(B^{\prime}(11, 0)\), so \(A^{\prime}B^{\prime} = \sqrt{25 + 4} = \sqrt{29}\). Length preserved.
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Under \(S\): \(A^{\prime}(4, 1)\), \(B^{\prime}(14, 3)\), so \(A^{\prime}B^{\prime} = \sqrt{100 + 4} = \sqrt{104}\). Length changed.
Rule \(R\), a translation, is rigid; rule \(S\) stretches horizontally and is not.
Common mistake
Watch out
Felix says a reflection is not a rigid transformation because the image is backward, so it cannot match the original.
Why it is wrong
Rigid means distances and angles are preserved, not that the figure faces the same way. A reflection keeps every length and angle, so the image is congruent. It only reverses orientation.
Related terms and lessons
- Translation (Geometry)
- Reflection (Geometry)
- Rotation (Geometry)
- Congruent Figures
- Congruent Triangles
- Dilation
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