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📐 Geometry · Transformations

Transformation tricks that make geometry move

Rotations, reflections, translations, and dilations — clarified.

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🔁 Transformations

Memory tricks

Proven mnemonics — fast to learn, hard to forget.

Four Transformation Types
TRRD (T=Translation/slide, R=Rotation/turn, R=Reflection/flip, D=Dilation/resize): Translation, Rotation, Reflection, Dilationion (scale)
Four Transformation Types
The four types of geometric transformations
Translation: add to coordinates (x+a, y+b). Rotation: turn around a fixed point. Reflection: flip over a line. Dilation: scale from a center. T, R, R preserve size (isometries). Dilation changes size.
T
Translation — slide
R
Rotation — turn
R
Reflection — flip
D
Dilation — scale
Difficulty: Beginner
Reflection Coordinate Rules
Reflect over x-axis: negate y → (x,-y). Over y-axis: negate x → (-x,y).
Reflection Coordinate Rules
Rules for reflecting across the axes and y=x
Over x-axis: (x,y) → (x,-y). Over y-axis: (x,y) → (-x,y). Over y=x: swap coordinates → (y,x). Over y=-x: swap and negate → (-y,-x).
Difficulty: Intermediate
Rotation Rules
90° counterclockwise: (x,y) → (-y,x). 90° clockwise: (x,y) → (y,-x).
Rotation Rules
Coordinate transformation rules for standard rotations
90° CCW: (x,y) → (-y,x). 90° CW: (x,y) → (y,-x). 180°: (x,y) → (-x,-y). 270° CCW = 90° CW.
Difficulty: Intermediate
Dilation Rules
Dilation from origin: multiply both coordinates by scale factor k
Dilation Rules
Multiply every coordinate by the scale factor
k > 1: enlargement. 0 < k < 1: reduction. k < 0: reduction + 180° rotation. Area scales by k² (scale factor squared).
Difficulty: Intermediate
Composition of Transformations
Composition of transformations: order matters — apply right to left
Composition of Transformations
Combining two or more transformations — sequence is critical
T₂∘T₁ means apply T₁ first, then T₂. Changing the order usually gives a different result. Two reflections over intersecting lines = a rotation. Two reflections over parallel lines = a translation.
Difficulty: Advanced
Isometries
Isometry: transformation that preserves distance and shape. Translation, rotation, reflection are isometries. Dilation is NOT.
Isometries
Which transformations preserve size — and which don't
Isometry (rigid motion): preserves distances between all points → preserves size and shape. Translations, rotations, and reflections are all isometries — the image is congruent to the pre-image. Dilation: NOT an isometry — changes size (unless scale factor = 1). Dilated image is similar but not congruent.
Difficulty: Intermediate
Glide Reflection
Glide reflection: translate then reflect over a line parallel to the translation direction
Glide Reflection
A composite transformation combining translation and reflection
Glide reflection = translation + reflection over a line parallel to the translation direction. The order doesn't matter here — same result either way. Footprints in snow: alternating left and right footprints form a glide reflection pattern. Classified as an isometry (preserves distance).
Difficulty: Advanced
Types of Symmetry
Symmetry: line symmetry (reflection maps figure to itself). Rotational symmetry (rotation maps to itself).
Types of Symmetry
Two ways a figure can be symmetric
Line (reflective) symmetry: one or more lines where reflection produces the same figure. Regular polygon with n sides has n lines of symmetry. Rotational symmetry: rotation of less than 360° maps figure to itself. Order of rotation: number of times it maps to itself in one full rotation. Regular hexagon: 6-fold rotational symmetry.
Difficulty: Intermediate
Tessellations
Tessellation: tiles fill a plane with no gaps or overlaps. Regular polygons that tessellate: equilateral triangle, square, regular hexagon.
Tessellations
Which regular polygons can tile the plane — and why
For a regular polygon to tessellate: interior angle must divide evenly into 360°. Equilateral triangle: 60° → 6 fit around a point ✓. Square: 90° → 4 fit ✓. Regular hexagon: 120° → 3 fit ✓. Regular pentagon: 108° → 360/108 = 3.33... ✗. Only these three regular polygons tessellate alone.
Difficulty: Intermediate
Dilation Scale Factors
Scale factor > 1: enlargement. Scale factor between 0 and 1: reduction. Scale factor = 1: identity (no change).
Dilation Scale Factors
Interpreting the scale factor of a dilation
Center of dilation: the fixed point from which the figure is scaled. Scale factor k > 1: image larger than pre-image. 0 < k < 1: image smaller. k = 1: identical image (identity transformation). k < 0: reduction AND 180° rotation. Image points: multiply each coordinate by k (if center is origin).
Difficulty: Intermediate
Properties of Rotations
Rotation center: each point travels the same angle on a circular arc centered at the rotation center
Properties of Rotations
What stays constant and what changes in a rotation
Every point moves through the same angle. The distance from each point to the center of rotation stays constant — points travel on circular arcs. The rotation center is the only point that stays fixed. Rotating 90° CCW four times = 360° = back to start. Composition of two reflections over intersecting lines = rotation.
Difficulty: Intermediate
Composition of Reflections
Two reflections over parallel lines = translation. Distance = twice the distance between the lines.
Composition of Reflections
How two reflections combine to create other transformations
Reflect over two parallel lines: result is a translation. Translation distance = 2× the distance between the lines. Direction: perpendicular to both lines. Reflect over two intersecting lines: result is a rotation. Rotation angle = 2× the angle between the lines. These relationships connect the four transformation types.
Difficulty: Advanced
Mnemonic
What it means
✅ 0❌ 0📚 0 left

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🎓 Common Exam Questions
Q: What does TRRD stand for and explain all four transformation types?
A: TRRD: Translation (slide): every point moves the same direction and distance. Rule: add vector (a,b) to all coordinates — (x,y) becomes (x+a, y+b). Isometry. Rotation (turn): every point rotates the same angle around a center point. Common rules about origin: 90 CCW: (-y,x), 180: (-x,-y), 90 CW: (y,-x), 270 CCW = 90 CW. Isometry. Reflection (flip): every point maps to its mirror image across a line of reflection. Isometry. Dilation (resize): every point moves toward or away from a center by a scale factor k. Rule from origin: (kx, ky). NOT an isometry — it changes size. Translation, rotation, and reflection are rigid motions (isometries) — the image is congruent to the preimage. Dilation produces a similar but not necessarily congruent figure.
Q: What are all the coordinate rules for reflections? Explain y=x and y=-x.
A: Reflection rules: Over x-axis: (x,y) to (x,-y) — negate y. Over y-axis: (x,y) to (-x,y) — negate x. Over origin: (x,y) to (-x,-y) — negate both (equivalent to 180 degree rotation). Over line y=x: (x,y) to (y,x) — swap x and y. Over line y=-x: (x,y) to (-y,-x) — swap x and y then negate both. How to remember y=x: the line y=x goes diagonally, reflecting over it swaps the coordinates. The line y=-x also goes diagonally but the other way, so swap AND negate.
Q: Explain composition of transformations with an example — why does order matter?
A: Composition: applying two or more transformations in sequence. The result depends on the ORDER — transformations are not commutative in general. Example: reflect (3,1) over x-axis, then translate by (2,0). Step 1: reflect: (3,1) to (3,-1). Step 2: translate: (3,-1) to (5,-1). Different order — translate first, then reflect: Step 1: (3,1) to (5,1). Step 2: (5,1) to (5,-1). Same result here! But other combinations would differ. Important result: two reflections over parallel lines = translation of twice the distance between lines. Two reflections over intersecting lines = rotation of twice the angle between lines.
Q: What is tessellation and which regular polygons tessellate?
A: Tessellation: a pattern of shapes that covers the plane with no gaps and no overlaps. For a regular polygon to tessellate alone, its interior angle must divide evenly into 360 degrees (since angles around a vertex must add to 360). Regular triangle: interior angle 60. 360/60 = 6 triangles per vertex. Works. Square: interior angle 90. 360/90 = 4 squares per vertex. Works. Regular hexagon: interior angle 120. 360/120 = 3 hexagons per vertex. Works. Regular pentagon: interior angle 108. 360/108 = 3.33 — not a whole number. Does not tessellate. Regular heptagon and higher: interior angles too large. Semi-regular tessellations combine different regular polygons — there are 8 such patterns.
Q: What is symmetry and what types does a square have?
A: Types of symmetry: Line symmetry (reflectional): a figure has line symmetry if a line exists such that reflecting the figure over it produces the same figure. The line is the line of symmetry or axis of symmetry. Rotational symmetry: a figure has rotational symmetry of order n if it maps to itself under a rotation of 360/n degrees. Order 1 = no rotational symmetry. Point symmetry: 180 degree rotational symmetry (maps to itself under 180 degree rotation). A square has: 4 lines of symmetry (horizontal, vertical, and two diagonals) and rotational symmetry of order 4 (maps to itself at 90, 180, 270, 360 degrees). A regular n-gon has n lines of symmetry and rotational symmetry of order n.