8.6 Transformations, Congruence & Similarity on the Coordinate Plane
Key Takeaways
- Translations, reflections, and rotations are rigid motions: they preserve every distance and every angle, so the image is congruent to the pre-image.
- Dilation is the one transformation that is not rigid - it preserves angles and shape but scales every length, 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 and a dilation does.
- Reflection reverses orientation while translation, rotation, and dilation preserve it, which is why a reflected figure reads backwards even though it is congruent.
- Coordinate rules must be applied in the stated order, because transformations do not generally commute - reflecting then rotating rarely lands where rotating then reflecting does.
8.6 Transformations, Congruence & Similarity on the Coordinate Plane
Quick Answer: A transformation maps every point $(x, y)$ of a pre-image to an image point. Translations, reflections, and rotations are rigid motions - they preserve every length and every angle, so pre-image and image are congruent. Dilation is not rigid: it preserves angles and shape but multiplies every length by the scale factor, so the image is similar. Grade 8 defines the two relationships by what maps one figure onto the other: congruent means a sequence of rigid motions does it; similar means a sequence of rigid motions and a dilation does it. Composite transformations must be applied in the stated order.
Where This Sits in the Blueprint
| Grade | Reporting category | Benchmark focus |
|---|---|---|
| Grade 8 | Geometric Reasoning (22-28%) | MA.8.GR.1 - transformations, congruence, and similarity, including single transformations and sequences applied to figures on the coordinate plane |
Geometric Reasoning is 22-28% of the grade 8 test, and transformations are the largest part of it. Scale factor arithmetic - how areas and volumes respond to scaling - is developed in § 8.5; this section is about the transformations themselves and what they preserve.
Rigid Motions
A transformation is a rigid motion (or isometry) when it preserves distance between points. Angles, side lengths, perimeter, and area all survive unchanged; only position, and sometimes orientation, differ.
1. Translation - a Slide
Every point moves the same horizontal distance $a$ and the same vertical distance $b$. Positive $a$ slides right, negative $a$ slides left; positive $b$ slides up, negative $b$ slides down.
Example. Translate $A(-3, 5)$ by $(x + 6,, y - 8)$: the image is $A'(3, -3)$.
Because every point moves identically, a translated figure keeps its exact size, shape, and orientation. Nothing turns and nothing flips.
2. Reflection - a Flip
| Line of reflection | Rule | What happens |
|---|---|---|
| $x$-axis | $(x, y) \rightarrow (x, -y)$ | $y$ changes sign |
| $y$-axis | $(x, y) \rightarrow (-x, y)$ | $x$ changes sign |
| $y = x$ | $(x, y) \rightarrow (y, x)$ | Coordinates swap |
| $y = -x$ | $(x, y) \rightarrow (-y, -x)$ | Swap and both change sign |
A reflection places each point the same perpendicular distance from the line of reflection, on the opposite side. Reflection is the only one of the four transformations that reverses orientation - a figure lettered clockwise comes back lettered counterclockwise. It is still congruent; it just reads backwards, the way a word does in a mirror.
3. Rotation - a Turn About the Origin
By convention, positive rotations run counterclockwise.
| Rotation | Rule | Equivalent |
|---|---|---|
| $90°$ counterclockwise | $(x, y) \rightarrow (-y,; x)$ | $270°$ clockwise |
| $180°$ | $(x, y) \rightarrow (-x,; -y)$ | Same either direction |
| $270°$ counterclockwise | $(x, y) \rightarrow (y,; -x)$ | $90°$ clockwise |
| $360°$ | $(x, y) \rightarrow (x,; y)$ | Back to start |
How to remember them without memorizing. Take $(3, 1)$, which sits in Quadrant I just above the $x$-axis. Rotate it $90°$ counterclockwise and it should land in Quadrant II, near the $y$-axis: $(-1, 3)$. That matches $(x, y) \rightarrow (-y, x)$. Test any doubtful rule on a single point in Quadrant I and check which quadrant the answer lands in.
Quadrant II | Quadrant I
(-, +) | (+, +)
| * (3, 1)
----------------+----------------
|
Quadrant III | Quadrant IV
(-, -) | (+, -)
90 deg CCW: (3, 1) -> (-1, 3) lands in Quadrant II [correct]
180 deg: (3, 1) -> (-3, -1) lands in Quadrant III [correct]
270 deg CCW: (3, 1) -> (1, -3) lands in Quadrant IV [correct]
The Non-Rigid Transformation: Dilation
A dilation centered at the origin with scale factor $k$ maps
| Property | Rigid motions | Dilation |
|---|---|---|
| Angle measures | Preserved | Preserved |
| Side lengths | Preserved | Multiplied by $ |
| Shape | Preserved | Preserved |
| Area | Preserved | Multiplied by $k^2$ |
| Relationship to pre-image | Congruent | Similar |
- $k > 1$ enlarges; $0 < k < 1$ reduces; $k = 1$ leaves the figure unchanged.
- A negative scale factor combines a dilation of magnitude $|k|$ with a $180°$ rotation about the origin, so the image lands on the opposite side of the center.
Congruence and Similarity, Defined by Transformation
This is the conceptual core of MA.8.GR.1, and it is what items actually test.
| Relationship | Definition | Notation |
|---|---|---|
| Congruent | A sequence of rigid motions maps one figure exactly onto the other | $\triangle ABC \cong \triangle DEF$ |
| Similar | A sequence of rigid motions and a dilation maps one figure onto the other | $\triangle ABC \sim \triangle DEF$ |
Read the definitions carefully and two facts fall out:
- Every congruent pair is also similar, with scale factor 1. Similarity is the broader category.
- A figure and its image under any single rigid motion are always congruent - no measuring required. If an item tells you a figure was reflected and then translated, the answer to "are they congruent?" is yes, before you look at a single coordinate.
Describing a sequence. Items often ask which sequence maps one figure onto another. Work it in this order:
1. Are the figures the SAME SIZE?
No -> a dilation is required; find k from one pair of corresponding sides
Yes -> rigid motions only
2. Is the ORIENTATION reversed (lettering runs the other way)?
Yes -> an odd number of reflections is involved
No -> translations, rotations, or an even number of reflections
3. Match one corresponding vertex pair to pin down the translation or the angle.
Composite Transformations: Order Matters
Apply transformations one at a time, in the order given, carrying the output of each step into the next.
Worked example. Point $P(4, -2)$ is reflected across the $y$-axis, then rotated $90°$ counterclockwise about the origin.
- Reflection across the $y$-axis: $(4, -2) \rightarrow (-4, -2)$.
- Rotation $90°$ CCW, $(x, y) \rightarrow (-y, x)$ applied to $(-4, -2)$: $\rightarrow (2, -4)$.
Final image: $P''(2, -4)$.
Now reverse the order. Rotate first, then reflect.
- Rotation $90°$ CCW: $(4, -2) \rightarrow (2, 4)$.
- Reflection across the $y$-axis: $(2, 4) \rightarrow (-2, 4)$.
Final image: $(-2, 4)$ - a different point. Transformations do not generally commute, which is exactly why items specify the order and why distractors are built from the reversed sequence.
| Sequence applied to $P(4, -2)$ | Step 1 | Step 2 | Result |
|---|---|---|---|
| Reflect over $y$-axis, then rotate $90°$ CCW | $(-4, -2)$ | $(2, -4)$ | $(2, -4)$ |
| Rotate $90°$ CCW, then reflect over $y$-axis | $(2, 4)$ | $(-2, 4)$ | $(-2, 4)$ |
[!IMPORTANT] Two reflections are not a wash. Reflecting across the $x$-axis and then the $y$-axis gives $(x, y) \rightarrow (x, -y) \rightarrow (-x, -y)$ - which is a $180°$ rotation. Orientation is reversed twice, so it comes back to normal, and the composite is a rotation rather than a reflection.
Common Exam Traps & Misconceptions
[!WARNING]
Trap 1: Confusing the $90°$ clockwise and counterclockwise rules
$90°$ counterclockwise is $(x, y) \rightarrow (-y, x)$; $90°$ clockwise is $(x, y) \rightarrow (y, -x)$. When unsure, rotate a single Quadrant I point and check which quadrant it should land in.
[!WARNING]
Trap 2: Applying a composite in the wrong order
"Reflected across the $x$-axis and then translated" means reflect first. Answer choices routinely include the result of the reversed order.
[!WARNING]
Trap 3: Believing a reflection changes size
A reflected figure is congruent. It looks different only because the orientation reversed; every length and angle is identical.
[!WARNING]
Trap 4: Calling a dilated figure congruent
A dilation with $k \ne 1$ produces a similar figure, not a congruent one. Angles match, lengths do not.
[!WARNING]
Trap 5: Forgetting that congruent figures are also similar
If an item asks whether two congruent triangles are similar, the answer is yes - similarity with a scale factor of 1.
Triangle ABC has vertex A located at coordinate (-3, 5). The triangle undergoes a 90° counterclockwise rotation about the origin, followed by a reflection across the x-axis. What are the coordinates of the final image point A''?
Triangle DEF is mapped onto triangle D'E'F' by a reflection across the y-axis followed by a translation 4 units down. Which statement about the two triangles is true?
Point M(-5, 2) is rotated 90 degrees counterclockwise about the origin and then reflected across the line y = x. What are the coordinates of the final image?