Transformations

GE.B.1Translations01/04

Translations — slide every point the same way.

Translation rule

(x, y) \to (x + a, y + b)

Step01of 04

A translation shifts every point of a figure by the same x and y amounts. No rotating, no flipping, no resizing — just a slide.

Notation: $T_{⟨ a, b ⟩}$ means "shift right $a$ , up $b$ ." Negative values shift left or down.

GE.B.1 — Translations — slide every point the same way.

Translation rule

(x, y) \to (x + a, y + b)

Step 01 of 04

A translation shifts every point of a figure by the same x and y amounts. No rotating, no flipping, no resizing — just a slide.

Notation: $T_{⟨ a, b ⟩}$ means "shift right $a$ , up $b$ ." Negative values shift left or down.

Step 02 of 04

Worked example. Translate triangle $A (- 3, - 1)$ , $B (- 1, - 1)$ , $C (- 2, 2)$ by $⟨ 5, 1 ⟩$ . Add 5 to each x, add 1 to each y:

A^{'} = (2, 0), B^{'} = (4, 0), C^{'} = (3, 3)

Step 03 of 04

Visualize. Pre-image (grey, dashed) and image (forest) are identical — same size, same shape, same orientation.

Each vertex moves the same vector

⟨ 5, 1 ⟩

Step 04 of 04

Properties translations preserve.

Side lengths	→unchanged
Angle measures	→unchanged
Orientation	→unchanged (no flipping)
Position	→CHANGED — that's the whole point

Key insight

Translations are the simplest rigid motion: every point obeys

(x, y) \to (x + a, y + b)

. Pre-image and image are congruent and identically oriented.

GE.B.2 — Reflections — flip across a line.

Step 01 of 04

A reflection flips a figure across a line of reflection (the mirror). Every point in the image is the same distance from the mirror as the original — just on the opposite side.

Across the x-axis	→ $(x, y) \to (x, - y)$
Across the y-axis	→ $(x, y) \to (- x, y)$
Across the line y = x	→ $(x, y) \to (y, x)$ (swap)
Across the origin	→ $(x, y) \to (- x, - y)$ (negate both)

Step 02 of 04

Worked example. Reflect $A (2, 1)$ , $B (4, 1)$ , $C (3, 4)$ across the y-axis. Negate every x:

A^{'} = (- 2, 1), B^{'} = (- 4, 1), C^{'} = (- 3, 4)

Reflection across the y-axis. Notice the orientation flips.

Step 03 of 04

Orientation flips. If the original triangle's vertices read clockwise, the reflected triangle's vertices read counter-clockwise. This is the key feature that distinguishes reflections from translations and rotations.

The line of reflection is the perpendicular bisector of every segment connecting a point to its image. That's the formal definition.

Step 04 of 04

What's preserved, what's not.

Side lengths	→unchanged
Angle measures	→unchanged
Orientation	→flipped
Position	→changed

Key insight

Reflection = mirror image. Use the substitution rule for the standard mirrors (axes, y = x), and the perpendicular-bisector definition for everything else. Orientation flips — that's the diagnostic.

GE.B.3 — Rotations — turn around a center.

Step 01 of 04

A rotation turns a figure around a fixed center by a specified angle. Positive angle = counter-clockwise (math convention). The most common rotations have memorable substitution rules.

$90°$ CCW about origin	→ $(x, y) \to (- y, x)$
$180°$ about origin	→ $(x, y) \to (- x, - y)$
$270°$ CCW about origin	→ $(x, y) \to (y, - x)$
$360°$	→identity (no change)

$90°$ CW = $270°$ CCW; same destination, same rule.

Step 02 of 04

Worked example. Rotate triangle $A (1, 1)$ , $B (4, 1)$ , $C (2, 3)$ by $90°$ CCW about the origin. Apply $(x, y) \to (- y, x)$ :

A^{'} = (- 1, 1), B^{'} = (- 1, 4), C^{'} = (- 3, 2)

90° counter-clockwise about the origin.

Step 03 of 04

About a non-origin center. Translate the center to the origin, rotate using the standard rule, translate back. Three steps, no new math.

To rotate $(x, y)$ by $90°$ CCW about $(h, k)$ :

(x, y) \to (h - (y - k), k + (x - h))

This is "subtract the center, apply the rotation rule, add the center back."

Step 04 of 04

What's preserved.

Side lengths	→unchanged
Angle measures	→unchanged
Orientation	→unchanged (rotation does NOT flip)
Position	→changed

Key insight

Three rigid motions: translate (slide), reflect (flip), rotate (turn). All three preserve size and shape, but only reflection flips orientation. Rotation rules are easy to memorize for multiples of 90° — for off-axis centers, "translate, rotate, translate back."

GE.B.4 — Compositions of transformations.

Step 01 of 04

A composition applies one transformation, then another. Notation: $g \circ f$ means "do $f$ first, then $g$ " — read RIGHT to LEFT.

Order matters. "Reflect then translate" usually doesn't equal "translate then reflect."

Step 02 of 04

Worked example. Reflect $A (1, 2)$ across the x-axis, then translate by $⟨ 3, 1 ⟩$ .

A (1, 2) reflect (1, - 2) translate (4, - 1)

Each step is a separate substitution. Easy to track when written out.

Step 03 of 04

Common compositions worth recognizing.

Two reflections across PARALLEL lines	→a translation (twice the distance between the lines)
Two reflections across INTERSECTING lines	→a rotation (twice the angle between the lines, about their intersection)
Reflection then translation along the line	→a "glide reflection" (a single named transformation)

These identities show that just the three basic rigid motions can build every other rigid motion through composition.

Step 04 of 04

The composition of any two rigid motions is itself a rigid motion. Side lengths and angles stay invariant through any sequence — so the final figure is always congruent to the original, no matter how many steps.

Key insight

Compositions read right-to-left:

g \circ f

means apply

f

first. Track each step separately. Two reflections often equal a single translation or rotation — useful for simplifying long chains.

GE.B.5 — Symmetry — line and rotational.

Step 01 of 04

A figure has symmetry if some transformation maps it onto itself. Two flavors matter at this level.

Line symmetry	→a reflection across some line maps the figure onto itself
Rotational symmetry	→a rotation by some angle (less than 360°) maps the figure onto itself

Step 02 of 04

Line symmetry — count the lines. A figure can have 0, 1, several, or infinitely many lines of symmetry.

Equilateral triangle	→3 lines (one through each vertex to the opposite midpoint)
Square	→4 lines (2 diagonals, 2 perpendicular bisectors of sides)
Regular hexagon	→6 lines
Regular n-gon	→n lines
Circle	→infinitely many (every diameter)
Scalene triangle	→0 lines

Step 03 of 04

Rotational symmetry — find the smallest angle. The order is how many distinct rotations (including the full $360°$ ) map the figure to itself.

Equilateral triangle	→order 3 — rotates onto itself every $120°$
Square	→order 4 — every $90°$
Regular hexagon	→order 6 — every $60°$
Regular n-gon	→order n — every $360°/ n$
Circle	→infinite order (any angle works)
Scalene triangle	→order 1 (only the trivial $360°$ )

Step 04 of 04

Quick test on a letter.

"A"	→1 line (vertical), no rotational
"H"	→2 lines (vertical + horizontal), order-2 rotational (180°)
"S"	→no lines, order-2 rotational
"O"	→infinite lines + infinite rotational (treated as a circle)
"R"	→no symmetry of either kind

Key insight

Line symmetry needs at least one mirror line. Rotational symmetry needs an angle < 360° that maps the figure to itself. A figure can have one kind, both kinds, or neither — every regular polygon has both.

Free diagnostic

Test your understanding: 10 questions, ~10 min.

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