8.5 Transformations in the Plane: Reflections, Translations, Rotations & Dilations
Key Takeaways
- Reflections, translations, and rotations are rigid motions that preserve both size and shape, so the image is always congruent to the original figure.
- A dilation multiplies every distance from a fixed center by a scale factor $k$, preserving shape and angle measure but changing size, so the image is similar rather than congruent.
- Two figures are congruent exactly when a sequence of rigid motions maps one onto the other, and similar exactly when a sequence of rigid motions plus one dilation does.
- Standard coordinate rules: reflection across the $x$-axis sends $(x, y)$ to $(x, -y)$, across the $y$-axis to $(-x, y)$, and a $180^\circ$ rotation about the origin to $(-x, -y)$.
- A dilation with $\lvert k \rvert > 1$ enlarges the figure, $0 < \lvert k \rvert < 1$ reduces it, and area always changes by the factor $k^2$.
Transformations in the Plane
Quick Summary: The blueprint's first Measurement/Geometry item asks you to "understand transformations in the plane, including reflections, translations, rotations, and dilations" and to "describe a sequence of transformations to demonstrate that one two-dimensional figure is either congruent or similar to a second." A transformation moves or resizes a figure. The pre-image is the original; the image is the result, usually labelled with primes such as $A'$ (read "A prime").
The Four Transformations at a Glance
| Transformation | Informal name | What changes | What is preserved | Result |
|---|---|---|---|---|
| Translation | Slide | Position | Size, shape, orientation | Congruent |
| Reflection | Flip | Position, orientation | Size, shape | Congruent |
| Rotation | Turn | Position, orientation | Size, shape | Congruent |
| Dilation | Resize | Position, size | Shape, angle measures | Similar |
The first three are called rigid motions (or isometries) because every distance is preserved. A dilation is not rigid: it scales distances.
Translations: Sliding a Figure
A translation shifts every point the same distance in the same direction. The rule is:
where $a$ is the horizontal shift (right if positive, left if negative) and $b$ is the vertical shift (up if positive, down if negative).
Example. Translate $A(2, -3)$ by 4 units left and 5 units up: $a = -4$, $b = +5$, so $A' = (2 - 4,\ -3 + 5) = (-2, 2)$.
Every point moves identically, so the image is the same size, the same shape, and faces the same way.
Reflections: Flipping Across a Line
A reflection flips the figure across a line of reflection, which acts as a mirror. Each point and its image are the same distance from that line, on opposite sides.
| Line of reflection | Rule | Effect |
|---|---|---|
| $x$-axis | $(x, y) \rightarrow (x, -y)$ | Negate the $y$-coordinate |
| $y$-axis | $(x, y) \rightarrow (-x, y)$ | Negate the $x$-coordinate |
| Line $y = x$ | $(x, y) \rightarrow (y, x)$ | Swap the coordinates |
| Line $y = -x$ | $(x, y) \rightarrow (-y, -x)$ | Swap and negate both |
Example. Reflect $B(5, 2)$ across the $x$-axis: $B' = (5, -2)$. Across the $y$-axis instead: $B' = (-5, 2)$.
A reflection reverses orientation: if the vertices of the pre-image read clockwise, the image reads counterclockwise. This is the feature that distinguishes a reflection from a rotation on a multiple-choice diagram.
Rotations: Turning About a Point
A rotation turns the figure about a fixed center of rotation through a given angle. Unless a problem says otherwise, rotations are counterclockwise about the origin.
| Rotation (counterclockwise about origin) | Rule |
|---|---|
| $90^\circ$ | $(x, y) \rightarrow (-y, x)$ |
| $180^\circ$ | $(x, y) \rightarrow (-x, -y)$ |
| $270^\circ$ | $(x, y) \rightarrow (y, -x)$ |
| $360^\circ$ | $(x, y) \rightarrow (x, y)$ — back to start |
Example. Rotate $C(3, 1)$ by $90^\circ$ counterclockwise about the origin: $C' = (-1, 3)$.
Direction matters. A $90^\circ$ clockwise rotation is the same as a $270^\circ$ counterclockwise rotation, so $(x, y) \rightarrow (y, -x)$. A $180^\circ$ rotation is the one case where direction is irrelevant — the result is identical either way.
Dilations: Resizing From a Center
A dilation multiplies the distance from a fixed center of dilation by a scale factor $k$. Centered at the origin, the rule is simply:
| Scale factor | Effect on the figure |
|---|---|
| $k > 1$ | Enlargement — the image is bigger |
| $k = 1$ | No change |
| $0 < k < 1$ | Reduction — the image is smaller |
| $k < 0$ | Resizes and rotates $180^\circ$ through the center |
Example. Dilate $D(6, -4)$ by $k = \frac{1}{2}$ about the origin: $D' = (3, -2)$. The image is half as far from the origin in every direction.
What a Dilation Preserves and What It Does Not
- Preserved: shape, angle measures, parallelism, and all ratios of lengths within the figure.
- Changed: every side length is multiplied by $k$; perimeter is multiplied by $k$; area is multiplied by $k^2$.
That last rule connects directly to the Area Ratio Principle in Section 8.4: if a photo is enlarged by a scale factor of 3, its area becomes $3^2 = 9$ times larger, not 3 times.
Describing a Sequence That Proves Congruence or Similarity
This is exactly what the blueprint asks for, and it turns the informal definitions from Section 8.4 into precise ones:
- Two figures are congruent if and only if some sequence of rigid motions (translations, reflections, rotations) maps one exactly onto the other.
- Two figures are similar if and only if some sequence of rigid motions together with a dilation maps one exactly onto the other.
Worked Example: Congruence
Triangle $ABC$ has vertices $A(1, 1)$, $B(4, 1)$, $C(1, 3)$. Triangle $A'B'C'$ has vertices $A'(1, -1)$, $B'(4, -1)$, $C'(1, -3)$.
Every $x$-coordinate is unchanged and every $y$-coordinate is negated, which is exactly the rule $(x, y) \rightarrow (x, -y)$. A single reflection across the $x$-axis maps the first triangle onto the second. Because a reflection is a rigid motion, the triangles are congruent.
Worked Example: Similarity
Triangle $PQR$ has vertices $P(1, 2)$, $Q(3, 2)$, $R(1, 5)$. Triangle $P'Q'R'$ has vertices $P'(2, 4)$, $Q'(6, 4)$, $R'(2, 10)$.
Each coordinate has been doubled, matching $(x, y) \rightarrow (2x, 2y)$. A dilation about the origin with scale factor $k = 2$ maps the first onto the second, so the triangles are similar with a ratio of 2. Their corresponding angles are equal, their side lengths are in the ratio $1 : 2$, and their areas are in the ratio $1 : 4$.
How to Answer These on the Test
- Compare one matched pair of coordinates and look for the pattern: a constant added means translation; a sign flip means reflection; coordinates swapped means rotation or a diagonal reflection; a common multiplier means dilation.
- Confirm the same rule works for every vertex — one matching pair is not enough.
- Decide congruent (no dilation used) or similar (a dilation was needed).
The trap to avoid: assuming any two same-shaped triangles are congruent. If a dilation with $k \neq 1$ appears anywhere in the sequence, the figures are similar but not congruent.
Point M(-3, 7) is reflected across the y-axis, and the image is then translated 5 units down. What are the final coordinates?
A rectangle with an area of 24 square units is dilated about the origin by a scale factor of k = 3. What is the area of the resulting image?
Triangle JKL has vertices J(2, 1), K(5, 1), and L(2, 6). Triangle J'K'L' has vertices J'(-1, 2), K'(-1, 5), and L'(-6, 2). Which single transformation maps JKL onto J'K'L', and what is the relationship between the triangles?
Which sequence of transformations would demonstrate that two triangles are similar but NOT congruent?