↔ · Symmetry & Transformations
The three motions that keep shape and size — slide, flip, turn.
Which letter looks the same in a mirror — A, or F?
Locate reflected points using equal distances from the mirror line.
Start with shapes and mirrors, then use coordinates as another representation.
Turn it however you like — a left glove never becomes a right one.
Open the immersive version →A reading path containing this lesson: Move, reflect and measure · 1/3
What stays the same after reflection?
Imagine folding paper so that the two pictures overlap exactly. The fold is a mirror line. A reflected point keeps its distance from that line on the other side.
A point two squares to the right is paired with a point two squares to the left. Perpendicular distances are equal, and points on the mirror line stay in place.
A reflected shape may face a different way, but its side lengths and angles stay the same. Reflection is different from enlargement.
Coordinates tell the same story: reflection in the y-axis sends (2,1) to (−2,1). Only the sign of the horizontal coordinate changes.
Moving every point three squares right is a translation, not a reflection. Lengths and angles are still preserved, but the rule pairing the points is different.
Read these symbols together
- ℓ
- A line that perpendicularly bisects the segment joining a point to its reflection.
- (x,y)
- The first coordinate gives horizontal position; the second gives vertical position.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — of the letters A and F, which looks the same in a mirror?
Move the L-block
Use the three buttons and sort what changes from what doesn't.
original
rotation 0°
Same shape does not mean same position
Congruence is another name for 'movable onto each other'.
Instead of measuring every side and angle, define sameness as 'overlaps after sliding, flipping, turning' — defining equality through motion is the root of modern geometry.
A flip borrows an extra dimension.
Flipping a plane figure requires lifting it into 3D. Left and right gloves, mirrored letters — flipped twins can never meet within their own plane.
Moving points and a fixed reference
What the block experiment confirmed, in mathematical language.
The three motions
Translation, reflection, rotation — all three preserve lengths and angles (shape and size unchanged).
Congruence
Two figures brought to perfect overlap by combining the motions — all corresponding sides and angles equal.
Line vs point symmetry
Two ways of overlapping yourself — A has line symmetry, N point symmetry, H both.
Make one more distinction
A flip can be reproduced by turning cleverly enough.
Not within the plane. A flipped L never overlaps the original however you rotate it — as a left glove never becomes a right glove by turning. Flipping requires 'lifting the paper': a fundamentally different motion.
Symmetry is one thing — matching left and right.
Folding onto itself (line symmetry) and turning 180° onto itself (point symmetry) differ. N fails the fold but survives the turn (point only); A is the reverse (line only); H does both.
Where else can this idea help?
Tiles and wallpaper
Every repeating pattern is 'one base piece + slides, flips, turns'. It's proven there are exactly 17 possible wallpaper patterns.
Nature's symmetries
Butterfly wings (line), starfish (rotational), snowflakes (6-fold) — the forms of life and crystals are classified in symmetry's language.
Car and aircraft design
Bodies are designed with left-right mirror symmetry, halving the work — draw half and reflection makes the rest. CAD's mirror tool IS the flip.
Game sprites
A character facing left isn't redrawn — the right-facing image is flipped. The classic memory-saving trick is a reflection.
Explain it in your own words
Compare with one possible explanation
Both preserve lengths and angles. Reflection pairs points across a line; translation moves all points by the same vector.
Try another question
Q1Of H, A, N, F — which have line symmetry?Show answer ▾
H and A (vertical axis). H also folds on a horizontal axis (two mirror lines) and survives a 180° turn — point symmetric too. N is point-only; F is neither.
Q2How many 90° clockwise turns return a figure to the start?Show answer ▾
Four (90°×4 = 360°). A flip returns in two — the same structure as (−1)×(−1)=+1 from the negatives lab.
Q3In how many ways does an equilateral triangle overlap itself?Show answer ▾
Six — three rotations (0°, 120°, 240°) and flips over three mirror lines. Counting a shape's self-overlaps leads to university group theory.
Try answering yourself before revealing it — getting it wrong is where learning starts.
The idea to take with you
Choose the reference and rule, then examine properties such as lengths and angles that stay unchanged.
What question comes next?
A reading path containing this lesson: Move, reflect and measure · 1/3
Previous concept
Polygons
Only after knowing sides and angles does 'unchanged under motion' become surprising.
← Polygons labLeads to next
Perimeter & Area
Motions that keep shape are mastered — now the two rulers for a shape's size: perimeter and area.
Go to the Perimeter & Area lab →Related lab
The Coordinate Plane
Write motion in numbers — (x,y)→(x+a,y+b) — and movement becomes computation.
Go to the The Coordinate Plane lab →Related lab