Concept 8 of 8Foundation

📌 Formula Sheet — Symmetry

Symmetry rules summarized — with the reasoning behind each.

Angle of rotational symmetry = 360° / order

Means: order = how many times it matches in a full turnUse when: finding smallest rotation that maps shape onto itself

Why: A full rotation is 360°. If a shape matches 'order' times, they must be evenly spaced → each spaced 360°/order apart.

⚠ Trap: Every shape has order at least 1 (full 360° match). True rotational symmetry is order ≥ 2.

Regular n-gon: n lines of symmetry AND rotational order = n

Means: equilateral ▲ = 3/3; square = 4/4; hexagon = 6/6Use when: figuring out symmetry of any regular polygon

Why: Regular polygons have identical sides and angles. Each vertex provides a symmetry axis, and each step from one vertex to the next is a valid rotation.

⚠ Trap: Non-regular polygons don't follow this rule. A rectangle has 2 lines (not 4) though it has 4 sides.

Reflect point (x, y) across X-AXIS → (x, −y)

Means: flip the y-coordinate's signUse when: reflections on a coordinate grid

Why: The x-axis is the 'mirror.' The point moves straight down (or up) by the same amount — distance from mirror stays same, but y goes negative.

⚠ Trap: Don't flip BOTH coordinates — that's rotation by 180° around origin, not reflection.

Reflect point (x, y) across Y-AXIS → (−x, y)

Means: flip the x-coordinate's signUse when: left-right mirror on grid

Why: The y-axis is vertical. The point moves horizontally by the same distance to the other side — x gets negated, y stays the same.

⚠ Trap: Remember which axis: reflection across X-axis flips Y (and vice-versa). The axis NOT being crossed is the one that keeps its sign.

Reflect point (x, y) across the line y = x → (y, x)

Means: swap the two coordinatesUse when: diagonal mirror

Why: The line y = x treats x and y as equals. Reflecting across it exchanges their roles — what was x becomes y, and y becomes x.

⚠ Trap: (5, 2) reflected across y = x is (2, 5), not (−5, −2).

Rotation 180° about origin: (x, y) → (−x, −y)

Means: half-turn sends each point to its "antipode"Use when: finding image after 180° rotation

Why: A half-turn flips direction in both x and y. Both coordinates negate.

⚠ Trap: This is NOT the same as reflecting across a single axis. It's two reflections combined (x-axis then y-axis).

💡 Tip:Reflection preserves SHAPE and SIZE. Only orientation (left/right) flips.

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