Geometric Transformations Cheat Sheet

A printable reference covering translations, reflections, rotations, dilations, coordinate rules, congruence, similarity, and composition for grades 7-10.

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Geometric transformations describe how figures move or change size on the coordinate plane. This cheat sheet helps students recognize translations, reflections, rotations, dilations, and compositions. It is useful for checking coordinate rules, comparing original figures with images, and deciding whether figures are congruent or similar. Students in grades 7-10 use these ideas in geometry proofs, graphing, and real-world design problems. The most important idea is that rigid transformations preserve size and shape, while dilations preserve shape but change size. Translations slide figures, reflections flip figures, and rotations turn figures around a center. Dilations use a scale factor $k$ to enlarge or reduce distances from a center. Compositions apply two or more transformations in order, so the sequence of steps matters.

Key Facts

A translation moves every point the same horizontal and vertical amount, using the rule $(x,y) \to (x+a,y+b)$ .
A reflection over the $x$ -axis uses the coordinate rule $(x,y) \to (x,-y)$ .
A reflection over the $y$ -axis uses the coordinate rule $(x,y) \to (-x,y)$ .
A rotation of $90^{\circ}$ counterclockwise about the origin uses the rule $(x,y) \to (-y,x)$ .
A rotation of $180^{\circ}$ about the origin uses the rule $(x,y) \to (-x,-y)$ .
A dilation centered at the origin with scale factor $k$ uses the rule $(x,y) \to (kx,ky)$ .
Rigid transformations preserve side lengths and angle measures, so the preimage and image are congruent.
Dilations preserve angle measures and multiply side lengths by $k$ , so the preimage and image are similar when $k>0$ .

Vocabulary

Transformation: A transformation is a rule that moves or changes a figure to create a new figure called the image.
Preimage: The preimage is the original figure before a transformation is applied.
Image: The image is the figure after a transformation is applied.
Rigid Transformation: A rigid transformation is a movement that preserves all distances and angle measures, such as a translation, reflection, or rotation.
Dilation: A dilation is a transformation that resizes a figure from a center using a scale factor $k$ .
Composition: A composition is a sequence of two or more transformations applied in a specific order.

Common Mistakes to Avoid

Mixing up reflection rules is wrong because reflecting over the $x$ -axis changes the sign of $y$ , while reflecting over the $y$ -axis changes the sign of $x$ .
Using the wrong rotation direction is wrong because $90^{\circ}$ counterclockwise uses $(x,y) \to (-y,x)$ , but $90^{\circ}$ clockwise uses $(x,y) \to (y,-x)$ .
Forgetting that translations move every point the same amount is wrong because the shape should slide without turning, flipping, stretching, or changing size.
Treating a dilation as a rigid transformation is wrong because a dilation with $k \ne 1$ changes side lengths even though it keeps angle measures the same.
Applying transformations in the wrong order is wrong because a composition can produce a different final image when the sequence changes.

Practice Questions

1 Triangle $ABC$ has vertices $A(1,2)$ , $B(4,2)$ , and $C(1,5)$ . Find the coordinates after the translation $(x,y) \to (x-3,y+1)$ .
2 Point $P(-2,5)$ is reflected over the $y$ -axis and then rotated $180^{\circ}$ about the origin. What are the final coordinates of $P$ ?
3 A rectangle has vertices $(0,0)$ , $(6,0)$ , $(6,3)$ , and $(0,3)$ . Dilate it from the origin by scale factor $k=\frac{1}{2}$ . What are the new vertices?
4 Explain why a reflection followed by a translation creates a congruent image, but a dilation with $k=2$ creates only a similar image.

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