Geometric Transformations
Grade 8 · Math · Study Guide
Move, flip, turn, and resize figures on the coordinate plane using four types of transformations.
A transformation changes the position, size, or orientation of a figure. The original is called the pre-image; the result is the image. The four main types are: translations (slides), reflections (flips), rotations (turns), and dilations (resize).
Translations slide a figure without changing its shape or orientation. Reflections flip a figure over a line of reflection, creating a mirror image. Rotations turn a figure around a center point by a given angle (90, 180, or 270 degrees). Dilations enlarge or shrink a figure by a scale factor centered at a point.
Translations, reflections, and rotations are rigid motions (isometries) that preserve size and shape — the figures are congruent. Dilations change size but preserve shape — the figures are similar.
| Type | Effect | Preserves Size? | Result |
|---|---|---|---|
| Translation | Slides the figure | Yes | Congruent |
| Reflection | Flips over a line | Yes | Congruent |
| Rotation | Turns around a point | Yes | Congruent |
| Dilation | Enlarges or shrinks | Only if k=1 | Similar |
Key terms
- Pre-image
- The original figure before a transformation.
- Image
- The resulting figure after a transformation; labeled with prime (A').
- Translation
- A slide — every point moves the same distance in the same direction.
- Reflection
- A flip over a line of reflection; creates a mirror image.
- Rotation
- A turn around a fixed center point by a given angle.
- Dilation
- An enlargement or reduction by a scale factor; preserves shape but not size.
- Rigid Motion
- Transformation preserving size and shape: translations, reflections, rotations.
Flashcards
- Translation rule (x+a, y+b)
- Move every point a units right/left and b units up/down.
- Reflection over x-axis
- (x, y) becomes (x, -y) — the y-coordinate changes sign.
- Reflection over y-axis
- (x, y) becomes (-x, y) — the x-coordinate changes sign.
- 90 degree rotation CCW
- (x, y) becomes (-y, x)
- 180 degree rotation
- (x, y) becomes (-x, -y)
- Dilation with scale factor k
- (x, y) becomes (kx, ky). k>1 enlarges; 0<k<1 shrinks.
Practice questions
Point A at (3, -2). After translation (x+4, y-1), where is A'?
- (7, -3)
- (7, -1)
- (-1, -3)
- (4, -2)
Show answer
Answer: (7, -3)
x: 3+4=7; y: -2+(-1)=-3. New point is (7, -3).
A reflection always produces a figure congruent to the original.
Show answer
Answer: True
Reflections are rigid motions — they preserve all distances and angles.
A dilation with scale factor greater than 1 creates a(n) ___ figure.
Show answer
Answer: enlarged
Scale factor greater than 1 multiplies all distances from the center, making the figure bigger.
Point B at (2, 5). After reflecting over x-axis, where is B'?
- (-2, 5)
- (2, -5)
- (-2, -5)
- (5, 2)
Show answer
Answer: (2, -5)
Reflecting over x-axis negates the y-coordinate: (2, 5) becomes (2, -5).
Which transformation changes the SIZE of a figure?
- Translation
- Reflection
- Rotation
- Dilation
Show answer
Answer: Dilation
Only dilation changes size. Translations, reflections, and rotations are rigid motions.
Fun facts
- Computer animation uses transformations constantly! Every time a character moves, software applies thousands of translations and rotations per second.
- Snowflakes have 6-fold rotational symmetry — they look identical after rotating 60, 120, 180, 240, or 300 degrees.
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