The four types of geometric transformations — an overview before diving into each one's specific rules.
The Mnemonic
Slide, Flip, Turn, Resize
A transformation takes a shape (the pre-image) and moves or changes it into a new shape (the image). There are exactly four basic types: Translation — slides a shape without turning or flipping it. Reflection — flips a shape over a line, creating a mirror image.
Rotation — turns a shape around a fixed point by a given angle. Dilation — resizes a shape larger or smaller, scaling from a fixed center point (this connects directly to Similar Figures — dilation IS what creates similar shapes).
Four transformations, four different effects on a shape's position, orientation, or size
💡 Memory Trick
"The four types of geometric transformations." Translation = SLIDE (straight-line move). Reflection = FLIP (mirror image). Rotation = TURN (around a point). Dilation = RESIZE (bigger or smaller). Three of these four keep the shape exactly the same size — only dilation changes size.
Why It Works
Size-Preserving vs. Size-Changing
Translations, reflections, and rotations are all called isometries (covered in depth in the next lesson) — they preserve every distance and angle in the shape, meaning the image is always CONGRUENT to the pre-image. Only the shape's position or orientation changes, never its actual size or proportions.
Dilation is fundamentally different — it's the one transformation that creates a SIMILAR (not congruent) image, scaling every distance by the same factor k, exactly as covered in Similar Figures. This is why dilation gets grouped with the other three as a "transformation" even though it behaves so differently: all four share the same basic structure (pre-image → rule → image), even though three preserve size and one doesn't.
Using It In A Proof
Identifying Which Transformation Occurred
Given a pre-image and its image, the first task is always identifying WHICH of the four transformations produced that specific change.
1
Check if the size changed
If the image is a different size than the pre-image, it's a dilation. If the size is identical, it's one of the other three.
2
Check orientation for size-preserving transformations
If the shape simply moved position without flipping or turning, it's a translation. If it appears mirror-flipped, it's a reflection. If it appears rotated around a point, it's a rotation.
3
Confirm using coordinate rules
Each transformation has a specific coordinate rule (previewed in Coordinate Transformations Summary, and covered in depth in the following lessons) that can confirm the identification precisely.
Full Worked Example
Identifying a Transformation From Before/After Coordinates
Given: Triangle ABC has vertices A(1,1), B(4,1), C(1,3). After a transformation, the image has vertices A'(2,2), B'(8,2), C'(2,6). Identify: which transformation occurred.
1
Compare side lengths to check for a size change
AB = 3 (original), A'B' = 6 (image) — the image is exactly twice as long on this side.
2
Recognize a size change means dilation
Since the size doubled, this must be a dilation, not a translation, reflection, or rotation (which all preserve size).
3
Confirm using the coordinate pattern
Each original coordinate was multiplied by 2: (1,1)→(2,2), (4,1)→(8,2), (1,3)→(2,6). This matches the dilation rule (x,y)→(kx,ky) with k=2.
This confirms both the transformation TYPE (dilation) and its specific scale factor (k=2) using the coordinate pattern from Coordinate Transformations Summary.
🎯 Quick Worked Example
A shape's image has the exact same size and shape as its pre-image, but appears mirror-flipped across a vertical line. Which transformation is this?
1
Check for a size change. None — the image is the same size, ruling out dilation.
2
Check the orientation change. Mirror-flipped, not simply slid or turned.
3
Conclude. This is a reflection.
📌 Exam Application
Exams often show a pre-image and image side by side and ask you to identify the transformation without giving explicit coordinates — practice recognizing the VISUAL signature of each type (slide vs. flip vs. turn vs. resize) so you can identify transformations at a glance, not just algebraically.
⚠️ Most Common Four Transformation Types Mistakes
Trap 1 — Assuming any change in position means translation: A reflection or rotation also changes a shape's position, but they additionally change its orientation (flipped or turned) — only a translation moves a shape with NO change in orientation at all.
Trap 2 — Forgetting dilation is the only size-changing transformation: If two shapes are different sizes, it must be a dilation (or a dilation combined with another transformation) — translations, reflections, and rotations alone can never produce a size difference.
✓ Quick Self-Test
1) Name the four basic transformation types. 2) Which of the four changes a shape's size? 3) What is the general term for the three size-preserving transformations? 4) A shape appears in a new position, same size, but turned 90° — which transformation is this? 5) A shape's image has coordinates exactly 3 times the pre-image's coordinates — which transformation is this, and what is its scale factor?