Transformations & similarity
Transformations: translations, reflections, rotations; similarity/scaling (enlargement); four-quadrant coordinates & distances.
A transformation is a rule that moves or resizes a shape. Some transformations slide, flip or turn a shape without changing its size — others grow or shrink it while keeping its exact proportions.
What it means
Picture a shape drawn on grid paper. A transformation takes every point of that shape and sends it somewhere new, following one consistent rule. The starting shape is the object; the result is the image.
Four transformations matter here:
- Translation — a slide. Every point moves the same distance in the same direction, e.g. “3 right and 2 up”. Nothing turns or flips; the shape looks identical, just relocated.
- Reflection — a flip across a mirror line. Each point lands the same distance on the opposite side of the line, so the image is a mirror copy.
- Rotation — a turn about a fixed centre by some angle (say 90° clockwise). The shape swings around like a hand on a clock.
- Enlargement (scaling) — a resize by a scale factor. A factor of 2 doubles every length; a factor of ½ halves them.
The first three are rigid: lengths and angles never change, so object and image are congruent (identical in size and shape). Enlargement changes size but not shape, giving a similar figure — same angles, all lengths multiplied by the same factor.
To describe where things go, we use the coordinate plane: two number lines, the horizontal x-axis and vertical y-axis, crossing at the origin (0, 0). They split the plane into four quadrants, so coordinates can be negative — the point (-3, 2) sits 3 left and 2 up. The distance between two points on the same horizontal or vertical line is just the difference of the coordinates that change.
Worked examples
Translation. Triangle corners at (1, 1), (4, 1), (1, 3). Slide “5 left, 4 down”: subtract 5 from each x, 4 from each y.
(1, 1) -> (-4, -3)
(4, 1) -> (-1, -3)
(1, 3) -> (-4, -1)The triangle keeps its shape and crosses into a new quadrant.
Reflection. Reflect (3, 2) across the y-axis: keep y, flip the sign of x -> (-3, 2). Across the x-axis instead: (3, -2).
Enlargement. A 2-by-3 rectangle enlarged by scale factor 3 becomes 6-by-9. Every side triples; the shapes are similar. Its area, though, grows by 3 × 3 = 9 times — from 6 to 54 square units.
Distance. From (-2, 5) to (4, 5): same height, so distance = 4 − (−2) = 6 units.
The generative-art connection
Transformations are the grammar of pattern, so you can literally watch them build art. Take one small motif and reflect it across two mirror lines, then rotate the pair — repeat, and a mandala or kaleidoscope appears. This is exactly what the Symmetry Artist tool does: you draw one stroke and the reflections and rotations paint the rest for you, in real time.
Our own dot-multiplier tool (https://dot-multiplier.nico.art) makes the point vividly — a single dot is copied by repeated rotation about a centre into a symmetric starburst. You are not decorating with symmetry; the rotation rule is the picture. Change the angle and the whole pattern reorganises. Enlargement, meanwhile, is the engine of self-similar art: shrink a shape by a scale factor and place copies inside itself and you get the beginnings of a fractal.
Common misconceptions
- “Rotation and reflection are the same.” A reflection flips handedness — a left hand becomes a right hand. A rotation never does; it only turns. Trace a scalene triangle and check whether the corners read clockwise or anticlockwise: reflection reverses that order, rotation keeps it.
- “Enlarging doubles the area when you double the sides.” Doubling every length multiplies area by 4, not 2 (
2 × 2). Scale factor acts on lengths; area scales by the factor squared. - “Negative coordinates are smaller numbers.”
(-5, 0)is further from the origin than(2, 0). The sign tells you direction, not size — distance is always positive.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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A triangle has corners at `(2, 1)`, `(5, 1)` and `(2, 3)`. Translate it "3 left and 4 up" (subtract `3` from each `x`, add `4` to each `y`). Give the three new corners, and say which quadrant the point that started at `(2, 1)` lands in.
Answer
`(2,1)->(-1,5)`, `(5,1)->(2,5)`, `(2,3)->(-1,7)`. The first new corner `(-1, 5)` is in the second quadrant (`x` negative, `y` positive). The triangle keeps its exact size and shape.
Art hook Canvas: draw the triangle, then on a key-press draw its translated copy in a second colour, plus faint arrows from each old corner to its new one so the equal-length, equal-direction slide is visible. Repeat the same slide several times to make a marching row of triangles.
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Reflect the point `(4, -2)` across the `y`-axis, then reflect that result across the `x`-axis. Give both answers, and describe the single transformation that gets you from `(4, -2)` straight to the final point.
Answer
Across `y`-axis (flip sign of `x`): `(-4, -2)`. Then across `x`-axis (flip sign of `y`): `(-4, 2)`. Going straight from `(4, -2)` to `(-4, 2)` flips both signs, which is a `180°` rotation about the origin.
Art hook Plot the original dot plus its two reflected images and the final `(-4, 2)` dot — one in each quadrant. Join the four dots to reveal they form a rectangle centred on the origin, a live demo that two perpendicular reflections make a `180°` rotation. Let the user drag the first dot and watch all four move in symmetry.
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A rectangle is `3` units wide and `5` units tall. Enlarge it by scale factor `2`. Give the new width, new height, and explain how many times bigger the area becomes.
Answer
New width `3 × 2 = 6`, new height `5 × 2 = 10`. Old area `= 15`, new area `= 60`, which is `4` times bigger. Area scales by the factor squared: `2 × 2 = 4`.
Art hook Draw the small rectangle then its enlargement nested at the same corner. Tile the big one with copies of the small one to *show* 4 fit inside — the area-scaling made visible as a grid of 4 mini-rectangles.
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On a four-quadrant grid, find the distance from `(-3, 4)` to `(5, 4)`, and the distance from `(-3, 4)` to `(-3, -6)`. Both pairs share a coordinate, so no diagonal is needed.
Answer
First pair share `y = 4`, so distance `= 5 − (−3) = 8` units (horizontal). Second pair share `x = −3`, so distance `= 4 − (−6) = 10` units (vertical). Distance is the difference of the coordinate that changes.
Art hook Number-line-on-a-grid: highlight the horizontal run in one colour and the vertical run in another, labelling each segment with its length. Build an L-shaped path and let the length update live as the user drags an endpoint.
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A right-angled triangle with a right angle, a `30°` angle and a `60°` angle is enlarged by scale factor `3`. What are the three angles of the enlarged triangle, and are the two triangles congruent, similar, or neither?
Answer
The angles are unchanged: `90°`, `30°`, `60°`. Enlargement multiplies lengths but never changes angles, so the triangles are **similar** (same shape, different size) but not congruent (sizes differ).
Art hook Draw a triangle then successively larger similar copies fanning out from one shared vertex, all angles preserved — a self-similar 'growth spiral' where only the size changes, colour-shifting by size.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Look at the point `(-4, 3)`. Without measuring, is it to the left or right of the `y`-axis, and above or below the `x`-axis? Which quadrant is it in?
Answer
`x = -4` is negative, so it is to the **left**; `y = 3` is positive, so it is **above**. That is the second quadrant.
Art hook Colour each of the four quadrants a different pastel and drop a labelled dot into the correct one as the user types coordinates — a live four-quadrant colour map.
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Plot these three points in order and join them: `(1, 1)`, `(1, -2)`, `(4, -2)`. What kind of angle is made at the corner `(1, -2)`?
Answer
The segment from `(1,1)` down to `(1,-2)` is vertical; from `(1,-2)` across to `(4,-2)` is horizontal. They meet at a right angle (`90°`).
Art hook Draw the two segments and, when they meet at a right angle, snap a little square corner-marker into place and flash the two segments green to celebrate the perpendicular meeting.
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Slide the point `(6, 2)` by "4 left and 5 down". Where does it land, and did its shape or size change at all?
Answer
`x: 6 − 4 = 2`, `y: 2 − 5 = −3`, so it lands at `(2, -3)`. A translation moves a shape without changing its size or shape at all.
Art hook Show a single dot and a ghost trail of the slide vector, then let the user press a key to slide the whole grid of dots by the same amount at once, forming a translated copy of the pattern.
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Reflect the point `(2, 5)` across the `x`-axis. Then, separately, reflect `(2, 5)` across the `y`-axis. Give both images and say which coordinate stayed the same each time.
Answer
Across the `x`-axis: flip `y` -> `(2, -5)` (the `x` stayed the same). Across the `y`-axis: flip `x` -> `(-2, 5)` (the `y` stayed the same).
Art hook Mirror-drawing tool: whatever the user draws in one quadrant is copied live across a chosen axis, so a butterfly's second wing paints itself.
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A square has corners at `(0, 0)`, `(2, 0)`, `(2, 2)`, `(0, 2)`. Enlarge it by scale factor `3` from the corner `(0, 0)`. Give the four new corners.
Answer
Multiply every coordinate by `3`: `(0,0)`, `(6,0)`, `(6,6)`, `(0,6)`. The square is now `6` by `6`.
Art hook Nest the enlargements sharing the bottom-left corner `(0,0)`: draw the original `2×2` square, then its scale-factor-`3` image as a `6×6` square in a fainter shade (optionally add the scale-factor-`2` `4×4` in between, labelled), so growth by scale factor shows as concentric squares.
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Rotate a scalene (all-sides-different) triangle `90°` clockwise about any centre point. Trace its corners in order: do they still read clockwise afterwards, or has the order reversed?
Answer
They still read clockwise — a rotation preserves the order/handedness of the corners no matter where the centre is; it only turns the shape. (A reflection would reverse the order.)
Art hook Place a scalene arrow-shape near a centre point and stamp copies at `90°, 180°, 270°` — four arrows pinwheeling around one point, all curling the same way, so you *see* handedness is preserved.
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Find the distance from `(3, -1)` to `(3, 6)` on the grid. (They share their `x`-coordinate, so it is a straight vertical distance.)
Answer
Same `x`, so distance `= 6 − (−1) = 7` units. Watch out: it is `7`, not `5` — you must count *across* zero.
Art hook Vertical number line across the origin: shade the run from `-1` up to `6` and print '7 units', updating as either endpoint is dragged, so crossing zero is felt visually.
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Spot the mistake. Sam says: "I enlarged a `4 cm` by `4 cm` square by scale factor `2`, so its area doubles from `16` to `32` square cm." What did Sam get wrong, and what is the true new area?
Answer
Sam scaled the area by `2`, but area scales by the factor **squared**. New side `= 8 cm`, new area `= 8 × 8 = 64` square cm — that is `2 × 2 = 4` times the original `16`, not double.
Art hook Split-screen: left shows the small square, right tiles the enlarged square with a `2×2` = 4-cell grid of the original, so the reader counts 4 copies and sees why the area quadruples.
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A flag shape is at corners `(1, 1)`, `(1, 4)`, `(3, 2)`. Translate it "2 right, 3 down", then reflect the result across the `x`-axis. Give the corners after each step.
Answer
After sliding (`x+2`, `y−3`): `(3,-2)`, `(3,1)`, `(5,-1)`. After reflecting across the `x`-axis (flip `y`): `(3,2)`, `(3,-1)`, `(5,1)`.
Art hook Animate the two-step move: the flag slides, pauses, then flips, leaving faint 'echo' copies of each stage so the whole choreography is visible as a little dance across the plane.
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Two triangles have the same three angles: `50°`, `60°`, `70°`, but one has sides twice as long as the other. Are they congruent, similar, or neither? Explain in one sentence.
Answer
**Similar** but not congruent: same angles and same shape, but the sizes differ (one is a scale-factor-`2` enlargement of the other). Congruent would need identical sizes too.
Art hook Draw a triangle and its scale-2 copy side by side, then overlay a colour-coded angle arc at each vertex to show the matching `50°/60°/70°` arcs are identical even though the triangles differ in size.
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Open challenge. Start with any small shape you like near the origin. Design a pattern using at least one reflection and at least one rotation. Describe the transformations you used and roughly what your finished pattern looks like.
Answer
Any valid design counts, e.g.: reflect the motif across the `y`-axis to make a mirror pair, then rotate the pair by `90°`, `180°`, `270°` about the origin to fill four directions — the result is a 4-fold symmetric mandala. Success = correctly names each transformation and its effect.
Art hook A mini kaleidoscope studio: the user draws one motif, toggles mirror axes and a rotation count (`4/6/8`), and the canvas paints the full symmetric pattern live — exactly the Symmetry-Artist / dot-multiplier idea in the page.