Transformations Translations Rotations Reflections Cheat Sheet

A printable reference covering translations, rotations, reflections, coordinate rules, congruence, and dilation for grades 6-8.

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This cheat sheet covers the main geometry transformations students use on the coordinate plane: translations, rotations, reflections, and dilations. It helps students recognize how a figure moves, write coordinate rules, and compare the original figure with its image. These skills are important for graphing, symmetry, congruence, similarity, and later geometry proofs. Rigid transformations keep the same size and shape, so translations, rotations, and reflections produce congruent figures. A translation slides a figure, a rotation turns it around a point, and a reflection flips it across a line. A dilation changes size by multiplying distances from a center by a scale factor, so it produces a similar figure rather than a congruent one.

Key Facts

A translation by $\langle a,b\rangle$ sends each point $\left(x,y\right)$ to $\left(x+a,y+b\right)$ .
A reflection across the $x$ -axis sends $\left(x,y\right)$ to $\left(x,-y\right)$ .
A reflection across the $y$ -axis sends $\left(x,y\right)$ to $\left(-x,y\right)$ .
A reflection across the line $y=x$ sends $\left(x,y\right)$ to $\left(y,x\right)$ .
A $90^{\circ}$ counterclockwise rotation about the origin sends $\left(x,y\right)$ to $\left(-y,x\right)$ .
A $180^{\circ}$ rotation about the origin sends $\left(x,y\right)$ to $\left(-x,-y\right)$ .
A $270^{\circ}$ counterclockwise rotation about the origin sends $\left(x,y\right)$ to $\left(y,-x\right)$ .
A dilation centered at the origin with scale factor $k$ sends $\left(x,y\right)$ to $\left(kx,ky\right)$ .

Vocabulary

Transformation: A transformation is a rule that moves or changes a figure to create an image.
Translation: A translation is a slide that moves every point the same distance and direction.
Rotation: A rotation is a turn around a fixed point by a given angle.
Reflection: A reflection is a flip across a line that creates a mirror image.
Rigid Transformation: A rigid transformation preserves side lengths and angle measures, so the image is congruent to the original figure.
Dilation: A dilation changes the size of a figure by a scale factor while keeping the same shape.

Common Mistakes to Avoid

Mixing up the coordinate rule for a $90^{\circ}$ rotation, because $\left(x,y\right)$ becomes $\left(-y,x\right)$ for counterclockwise rotation about the origin, not $\left(y,-x\right)$ .
Adding the translation values to the wrong coordinates, because $\langle a,b\rangle$ means add $a$ to $x$ and add $b$ to $y$ .
Reflecting over the wrong axis, because reflection across the $x$ -axis changes the sign of $y$ , while reflection across the $y$ -axis changes the sign of $x$ .
Calling every transformation congruent, because dilations change side lengths when $k \neq 1$ and usually make similar figures instead.
Forgetting to transform every vertex, because the image of a polygon is found by applying the same rule to each point.

Practice Questions

1 Translate point $A\left(-2,5\right)$ by $\langle 4,-3\rangle$ . What are the coordinates of $A'$ ?
2 Rotate point $B\left(3,-1\right)$ $90^{\circ}$ counterclockwise about the origin. What are the coordinates of $B'$ ?
3 Reflect point $C\left(-6,2\right)$ across the $y$ -axis, then dilate the result by scale factor $\frac{1}{2}$ centered at the origin. What are the final coordinates?
4 A triangle is reflected across the $x$ -axis and then translated $5$ units right. Explain whether the final triangle is congruent to the original and why.

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