8.6 Transformations, Congruence & Similarity on the Coordinate Plane

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

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

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

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

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.

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

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

• $k > 1$ 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.

Read the definitions carefully and two facts fall out:

1. Every congruent pair is also similar, with scale factor 1. Similarity is the broader category.
2. 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.

1. Reflection across the $y$-axis: $(4, -2) \rightarrow (-4, -2)$.
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.

1. Rotation $90°$ CCW: $(4, -2) \rightarrow (2, 4)$.
2. 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.

---

Common Exam Traps & Misconceptions