8.5 Transformations in the Plane: Reflections, Translations, Rotations & Dilations

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

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:

$$(x, y) \rightarrow (x + a,\ y + b)$$

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.

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.

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:

$$(x, y) \rightarrow (kx,\ ky)$$

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

1. 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.
2. Confirm the same rule works for every vertex — one matching pair is not enough.
3. 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.