Rigid Transformation

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Rigid Transformation: meaning, method and solved example (Geometry) – Maths PDF

In short: A rigid transformation repositions a shape by sliding, flipping or turning it, alone or in a chain, so that every length and every angle measure survives intact.

GeometryGeometryG-CO.A.5

What is Rigid Transformation?

A rigid transformation relocates a shape without bending, stretching or shrinking it. Slides (translations), flips (reflections) and turns (rotations) all qualify, and so does any chain of them.

Because distances and angle measures survive, the resulting image is congruent to the original. Points keep their order and parallel sides stay parallel; only position or orientation can change. Mathematicians also call such a move an isometry.

How to spot it

  • A problem says a shape is slid, flipped or turned and asks whether the new shape is the same size.
  • You must show two figures are congruent by describing a sequence of moves that maps one onto the other.
  • Coordinates of vertices are given before and after, and side lengths are to be compared.
  • The wording mentions a mirror line, a center of rotation or a translation vector.

The method

  1. Identify each move: a slide along a vector, a flip over a line, or a turn about a point.
  2. Apply the moves one at a time to every vertex, recording the new coordinates.
  3. Compare side lengths and angle measures of the image with the original.
  4. If every length and angle matches, the move was rigid and the figures are congruent.
PQRP2Q2R2PQ = 4, QR = 3, PR = 5image: same lengths 4, 3, 5
A reflection then a slide down 3 moves the blue triangle to the red one without changing any side.

Rigid Transformation: solved example

Problem

Triangle \(PQR\) has vertices \(P(2, 1)\), \(Q(6, 1)\) and \(R(6, 4)\). It is reflected across the y-axis and then translated 3 units down. Find the final vertices and verify that the side lengths are unchanged.

Solution

Reflection across the y-axis changes \((x, y)\) to \((-x, y)\): \((-2, 1)\), \((-6, 1)\), \((-6, 4)\). Moving 3 down subtracts 3 from each y: \(P_2(-2, -2)\), \(Q_2(-6, -2)\), \(R_2(-6, 1)\).

Lengths: \(PQ = 4 = P_2Q_2\), \(QR = 3 = Q_2R_2\), and \(PR = \sqrt{16 + 9} = 5 = P_2R_2\). Nothing changed except position, so the image is congruent to the original.

Try it

Question 1

A map is enlarged on a copier so that every distance doubles. Is that a rigid move?

Show the answer

No. Lengths change, so the copy is not congruent to the original, although its shape stays similar.

Question 2

A segment from \((1, 2)\) to \((4, 6)\) is translated 5 units right. What is the length of the image?

Show the answer

The length stays \(\sqrt{3^2 + 4^2} = 5\), because sliding is rigid.

Rigid Transformation vs. Dilation

Dilation scales a figure about a center, so lengths change and only the shape survives. A rigid transformation also keeps lengths. Slide a 6-inch ruler across a desk and it stays 6 inches; photocopy it at 150 percent and it becomes 9 inches.

Dilation

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