Transformations & Symmetry Cheat Sheet

A printable reference covering translations, reflections, rotations, dilations, symmetry, and coordinate rules for grades 8-10.

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Transformations describe how figures move or change on the coordinate plane while keeping track of their points. This cheat sheet helps students compare translations, reflections, rotations, and dilations using clear coordinate rules. It is useful for graphing images, identifying congruence and similarity, and recognizing symmetry in geometric figures. Students in grades 8 through 10 need these tools for proofs, coordinate geometry, and problem solving. The main ideas are that rigid transformations preserve size and shape, while dilations preserve shape but change size. Coordinate rules such as $(x,y) \to (x+a,y+b)$ or $(x,y) \to (-x,y)$ show exactly how each point moves. Symmetry occurs when a figure maps onto itself after a reflection or rotation. Understanding the difference between congruent and similar images is the key to using transformations correctly.

Key Facts

A translation moves every point the same horizontal and vertical distance using the rule $(x,y) \to (x+a,y+b)$ .
A reflection over the $x$ -axis uses the rule $(x,y) \to (x,-y)$ .
A reflection over the $y$ -axis uses the 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 $(x,y) \to (kx,ky)$ .
Translations, reflections, and rotations are rigid transformations because they preserve side lengths and angle measures.
A figure has line symmetry if a reflection across a line maps the figure exactly onto itself.

Vocabulary

Transformation: A transformation is a rule that moves or changes a figure to create an image.
Preimage: The preimage is the original figure before a transformation is applied.
Image: The image is the new figure after a transformation is applied.
Rigid Transformation: A rigid transformation preserves distance and angle measure, so the image is congruent to the preimage.
Dilation: A dilation enlarges or reduces a figure by multiplying distances from a center by a scale factor.
Symmetry: Symmetry occurs when a figure can be transformed onto itself by a reflection or rotation.

Common Mistakes to Avoid

Mixing up the reflection rules for the axes is common because students often change the wrong coordinate. For a reflection over the $x$ -axis, only $y$ changes sign, so $(x,y) \to (x,-y)$ .
Using the wrong rotation direction gives an incorrect image. A $90^{\circ}$ counterclockwise rotation about the origin is $(x,y) \to (-y,x)$ , while a $90^{\circ}$ clockwise rotation is $(x,y) \to (y,-x)$ .
Forgetting to multiply both coordinates in a dilation changes the shape incorrectly. A dilation centered at the origin with scale factor $k$ must use $(x,y) \to (kx,ky)$ .
Calling every transformed figure congruent is wrong because dilations can change size. Translations, reflections, and rotations preserve congruence, but dilations with $k \ne 1$ create similar figures.
Assuming rotational symmetry means any rotation works is incorrect. A figure has rotational symmetry only if a rotation less than $360^{\circ}$ maps it exactly onto itself.

Practice Questions

1 Translate point $A(3,-2)$ using the rule $(x,y) \to (x-4,y+5)$ . What are the coordinates of $A'$ ?
2 Reflect point $B(-6,4)$ over the $y$ -axis. What are the coordinates of $B'$ ?
3 Rotate point $C(2,7)$ $90^{\circ}$ counterclockwise about the origin, then dilate the result by scale factor $k=2$ centered at the origin. What are the final coordinates?
4 A triangle is reflected over a line and then translated. Explain why the final triangle must be congruent to the original triangle.