Geometry Translations, Reflections and Rotations Lesson (Free PDF)

Geometry math lesson: Translations, Reflections and Rotations – Maths PDF

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This geometry transformations lesson for Geometry covers translations, reflections and rotations (Common Core G-CO.A.2, G-CO.A.4, G-CO.A.5), with a free PDF to print.

A transformation maps each point of a figure to a new point. Translations slide a figure, reflections flip it across a line, and rotations turn it about a center. These three rigid motions keep lengths and angle measures the same. Each has a coordinate rule.

Geometry · Common Core State Standards: G-CO.A.2, G-CO.A.4, G-CO.A.5

Key facts in this geometry transformations lesson

Key facts

A translation moves every point the same distance in the same direction: (x, y) to (x + a, y + b).
A reflection across the x-axis sends (x, y) to (x, -y); across the y-axis it sends (x, y) to (-x, y).
A 90 degree counterclockwise rotation about the origin sends (x, y) to (-y, x).
Rigid motions preserve distance and angle measure, so the image is congruent to the preimage.
When transformations are combined, the order matters.

What is a transformation?

A transformation takes each point of a figure, called the preimage, and maps it to a new point. The new figure is the image. We name image points with a prime mark, so A becomes A′. A transformation that keeps all distances and angle measures the same is a rigid motion.

How does a translation work?

A translation slides every point the same distance in the same direction. On the coordinate plane, the rule is \((x, y) \text{ maps to } (x + a, y + b)\).

Translation: every point moves 5 right and 2 up.

How does a reflection work?

A reflection flips a figure across a line of reflection. The line is the perpendicular bisector of every segment that joins a point to its image.

Reflection across the x-axis: (x, y) goes to (x, -y).

Reflection across	Rule
the x-axis	\((x, y) \text{ maps to } (x, -y)\)
the y-axis	\((x, y) \text{ maps to } (-x, y)\)
the line y = x	\((x, y) \text{ maps to } (y, x)\)

How does a rotation work?

A rotation turns a figure about a fixed point, the center, by a given angle. A positive angle means counterclockwise. Every point stays the same distance from the center.

Rotation of 90° counterclockwise about the origin: (x, y) goes to (-y, x).

Rotation about the origin	Rule
90° counterclockwise	\((x, y) \text{ maps to } (-y, x)\)
180°	\((x, y) \text{ maps to } (-x, -y)\)
270° counterclockwise	\((x, y) \text{ maps to } (y, -x)\)

How do you combine transformations?

You can apply one transformation to the image of another. The order can change the result, so always work in the order given.

A reflection followed by a translation is a sequence of rigid motions.

Example: Rotate \(P(3, -2)\) by 90° counterclockwise about the origin, then translate by \((x + 4, y)\). The rotation gives \((2, 3)\), and the translation gives \((6, 3)\).

Geometry transformations lesson: questions and answers

What is the difference between a preimage and an image?

The preimage is the original figure and the image is the figure after the transformation. Image points are named with a prime, so the image of A is A prime.

Which transformations are rigid motions?

Translations, reflections and rotations are rigid motions because they preserve distance and angle measure. A dilation is not a rigid motion because it changes lengths.

How do you reflect a point across the y-axis?

Change the sign of the x-coordinate and keep the y-coordinate. The point (4, 6) reflects to (-4, 6).

Which way is a positive rotation?

Counterclockwise is positive. A rotation of 90 degrees clockwise is the same as a rotation of 270 degrees counterclockwise.

Does the order of two transformations matter?

Often yes. Translating (3, 4) by (x + 2, y – 1) and then reflecting across the x-axis gives (5, -3), but reflecting first gives (5, -5).

How to use this geometry transformations lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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