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 sends each point to .
- A reflection across the -axis sends to .
- A reflection across the -axis sends to .
- A reflection across the line sends to .
- A counterclockwise rotation about the origin sends to .
- A rotation about the origin sends to .
- A counterclockwise rotation about the origin sends to .
- A dilation centered at the origin with scale factor sends to .
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 rotation, because becomes for counterclockwise rotation about the origin, not .
- Adding the translation values to the wrong coordinates, because means add to and add to .
- Reflecting over the wrong axis, because reflection across the -axis changes the sign of , while reflection across the -axis changes the sign of .
- Calling every transformation congruent, because dilations change side lengths when 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 by . What are the coordinates of ?
- 2 Rotate point counterclockwise about the origin. What are the coordinates of ?
- 3 Reflect point across the -axis, then dilate the result by scale factor centered at the origin. What are the final coordinates?
- 4 A triangle is reflected across the -axis and then translated units right. Explain whether the final triangle is congruent to the original and why.
Understanding Transformations Translations Rotations Reflections
A transformation is best understood as a rule that matches every original point with one new point. Labeling vertices helps you track that match. If point A becomes point A prime, every feature connected to A must move with it.
For a rigid movement, side lengths stay equal, angle measures stay equal, parallel lines remain parallel, and area stays the same. The order of the vertices matters too. A slide or turn keeps the clockwise or counterclockwise order of the labels.
A flip reverses that order. This reversal is a useful clue when two pictures look almost identical.
Coordinate rules are shortcuts, but they work because of movement on the grid. Before applying a rule, identify what stays fixed. In a translation, every point travels the same horizontal distance and the same vertical distance.
In a rotation, the center is fixed. Each point stays the same distance from that center while turning through the same angle. Draw segments from the center to a few vertices when a rotation feels confusing.
A turn around a point other than the origin can be handled in three stages. Move the center to the origin, rotate the figure, then move everything back.
Reflections depend on a line of symmetry, not merely on a direction. The mirror line is exactly halfway between each point and its image. The segment joining a point to its image meets the mirror line at a right angle.
This fact helps with reflections across vertical, horizontal, or diagonal lines that are not the usual axes. Dilations require similar care with the center. A point, its image, and the center lie on one straight line.
A positive scale factor places the image on the same ray from the center. A negative scale factor puts it on the opposite side, which combines resizing with a half turn.
Many problems use more than one movement. The order can change the final image. Sliding a shape before turning it usually gives a different result from turning it before sliding it.
Check your work by graphing at least two vertices, then compare the expected side lengths, directions, and location. These ideas appear in tiled floors, logos, computer graphics, maps, camera rotations, and patterns in art.
When studying, do not memorize a list without a sketch. Say what happens to the horizontal position and vertical position, mark the fixed point or line, then verify the image with the properties that must remain true.