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Rigid
Transformations

Master translations, reflections, and rotations: learn how to perform each one and sequence them to map any figure onto a congruent one.

Geometry

What Are Rigid Transformations?

A rigid transformation (also called an isometry) moves every point of a figure to a new location without changing its size or shape. Distances and angle measures are preserved exactly.

Translation
Slides every point the same direction and distance.
Reflection
Flips the figure over a line of reflection.
Rotation
Turns the figure around a fixed center point.

Translation: Sliding a Figure

A translation maps every point \( (x, y) \) to \( (x + a,\; y + b) \), where the translation vector is \( \langle a, b \rangle \). Every point moves the same direction and distance.

A B C vector A' B' C'

Reflection: Flipping a Figure

A reflection maps every point to its mirror image across a line of reflection. Each point and its image are equidistant from the line.

line of reflection P Q R P' Q' R'

Rotation: Turning a Figure

A rotation turns every point through a given angle about a fixed center of rotation. Positive angles go counterclockwise; negative angles go clockwise.

O
\(90^\circ\)
 A B C A' B' C'

The Isometry Guarantee

This means if \( \triangle ABC \) has sides of length 3, 4, 5 and angles \( 37^\circ, 53^\circ, 90^\circ \), then after any rigid transformation the image \( \triangle A'B'C' \) has exactly the same measurements.

Preserved
Side lengths, angle measures, parallelism, betweenness, area.
Not Preserved
Position (always changes). Orientation may change (reflections reverse it).

Composing Sequences of Transformations

You can chain two or more rigid transformations into a composition. Apply the first transformation, then apply the next one to the result. Order matters!

pre-image reflect after step 1 translate after step 2 rotate image

Rigid vs. Non-Rigid Transformations

Rigid (Isometry) Non-Rigid
Examples Translation, Reflection, Rotation Dilation, Stretch, Shear
Distance Preserved Changed
Angle Measure Preserved May change (shear)
Congruence Image ≅ Pre-image Image ~ or ≇ Pre-image
Shape Same Similar or different

Why Order Matters

Transformations are generally not commutative. Performing a reflection then a translation often produces a different image than the same translation then the reflection.

reflection line
Reflect, then Translate by \( \langle 40, -50 \rangle \)
 start result A
Translate by \( \langle 40, -50 \rangle \), then Reflect
 result B Result A ≠ Result B

Step-by-Step: Performing a Transformation

1
Identify the transformation type: Is it a translation, reflection, or rotation?
2
Note the parameters: Vector \( \langle a, b \rangle \), line of reflection, or center + angle of rotation.
3
Apply the rule to each vertex of the figure. Use coordinate rules or compass/straightedge.
4
Connect the image points in the same order to draw the transformed figure.
5
Verify: Check that all distances and angles match the original (measure or compute).

Example 1: Translation

Translate \( \triangle DEF \) with vertices \( D(1, 2) \), \( E(4, 2) \), \( F(4, 5) \) by the vector \( \langle 3, -1 \rangle \).

0 1234567 12345 D(1,2) E(4,2) F(4,5) D'(4,1) E'(7,1) F'(7,4)

Example 2: Reflection over the y-axis

Reflect \( \triangle GHI \) with vertices \( G(2, 1) \), \( H(5, 1) \), \( I(5, 4) \) over the y-axis.

y-axis 12345 -1-2-3-4-5 1234 G H I G' H' I'

Example 3: Rotation of \( 90^\circ \) About the Origin

Rotate \( \triangle JKL \) with \( J(1, 2) \), \( K(4, 2) \), \( L(4, 5) \) by \( 90^\circ \) counterclockwise about the origin.

O 12345 -1-2-3-4-5 12345 J(1,2) K(4,2) L(4,5) J'(-2,1) K'(-2,4) L'(-5,4)
\(90^\circ\)

Example 4: Sequencing Two Transformations

Take point \( A(3, 1) \). First reflect over the x-axis, then translate by \( \langle -4, 3 \rangle \). Where does \( A \) end up?

1
Reflect over x-axis: \( (x, y) \to (x, -y) \). So \( A(3, 1) \to A'(3, -1) \).
2
Translate by \( \langle -4, 3 \rangle \): add to the result. \( A'(3, -1) \to A''(3 + (-4),\; -1 + 3) = A''(-1, 2) \).

Common Mistakes

✕ Wrong Order
Applying transformations in the wrong sequence. Always follow the stated order: first transformation first.
✕ Sign Errors
Forgetting to negate coordinates. For a \( 90^\circ \) rotation, \( (x,y) \to (-y, x) \): do not drop the negative.
✕ Wrong Reflection Line
Reflecting over the x-axis when the problem says y-axis (or vice versa). Read the line of reflection carefully.

Summary of Coordinate Rules

Transformation Rule Key Detail
Translation by \( \langle a, b \rangle \) \( (x+a,\; y+b) \) Direction + distance
Reflect over x-axis \( (x, -y) \) Negate y
Reflect over y-axis \( (-x, y) \) Negate x
Reflect over \( y = x \) \( (y, x) \) Swap coordinates
Rotate \( 90^\circ \) CCW \( (-y, x) \) Negate old y
Rotate \( 180^\circ \) \( (-x, -y) \) Negate both
Rotate \( 270^\circ \) CCW \( (y, -x) \) Negate old x

Question 1 of 5

What is the image of point \( (3, -2) \) after a reflection over the y-axis?

Question 2 of 5

Point \( P(5, 2) \) is rotated \( 90^\circ \) counterclockwise about the origin. What are the coordinates of \( P' \)?

Question 3 of 5

Which transformation is not a rigid transformation?

Question 4 of 5

Point \( A(1, 4) \) is first reflected over the x-axis, then translated by \( \langle 2, 3 \rangle \). What is the final image \( A'' \)?

Question 5 of 5

Triangle \( \triangle XYZ \) has vertices \( X(0, 0) \), \( Y(4, 0) \), \( Z(0, 3) \). After a \( 180^\circ \) rotation about the origin, what are the vertices of the image?