Geometry: Transformations

8.F.1Translations01/04

Translations.

Translation rule

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

Step01of 04

A translation slides every point of a figure by the same amount. Direction and distance, no rotation, no flipping, no resizing.

Notation $T_{⟨ a, b ⟩}$ means "shift right $a$ , up $b$ ."

8.F.1 — Translations.

Translation rule

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

Step 01 of 04

A translation slides every point of a figure by the same amount. Direction and distance, no rotation, no flipping, no resizing.

Notation $T_{⟨ a, b ⟩}$ means "shift right $a$ , up $b$ ."

Step 02 of 04

Worked example. Translate triangle $A (1, 2)$ , $B (4, 2)$ , $C (2, 5)$ by $⟨ 3, - 1 ⟩$ .

A^{'} = (4, 1), B^{'} = (7, 1), C^{'} = (5, 4)

Step 03 of 04

Pre-image and image are CONGRUENT (same size + same shape) and have the SAME ORIENTATION (no flipping). Side lengths and angles preserved.

Step 04 of 04

Translation describes a vector — distance and direction. Two figures related by a translation are congruent and identically oriented.

Key insight

Translation = slide. Add the same vector to every vertex's coordinates. Side lengths, angles, and orientation all stay the same.

8.F.2 — Reflections.

Step 01 of 04

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

Across the x-axis	→ $(x, y) \to (x, - y)$
Across the y-axis	→ $(x, y) \to (- x, y)$
Across line y = x	→ $(x, y) \to (y, x)$

Step 02 of 04

Worked example. Reflect $A (2, 1), B (5, 1), C (3, 4)$ across the x-axis. Negate every y:

A^{'} = (2, - 1), B^{'} = (5, - 1), C^{'} = (3, - 4)

Step 03 of 04

Orientation flips. If the original triangle's vertices read clockwise, the reflected one reads counter-clockwise. This is the diagnostic that distinguishes reflection from translation/rotation.

Step 04 of 04

Side lengths and angle measures are preserved (it's still congruent), but ORIENTATION is reversed. The line of reflection is the perpendicular bisector of every segment connecting a point to its image.

Key insight

Reflection = mirror image. Standard mirrors (axes, y = x) have memorable substitution rules. Orientation flips — that's the test. Pre-image and image are congruent.

8.F.3 — Rotations.

Step 01 of 04

A rotation turns a figure around a fixed point (the center of rotation) by a specified angle. Counter-clockwise = positive angle (math convention).

$90°$ CCW about origin	→ $(x, y) \to (- y, x)$
$180°$ about origin	→ $(x, y) \to (- x, - y)$
$270°$ CCW about origin	→ $(x, y) \to (y, - x)$

Step 02 of 04

Worked example. Rotate $A (1, 2), B (4, 2), C (2, 5)$ by $90°$ CCW about the origin.

A^{'} = (- 2, 1), B^{'} = (- 2, 4), C^{'} = (- 5, 2)

Step 03 of 04

$90°$ CW = $270°$ CCW = same destination. $360°$ brings the figure back to itself (identity rotation).

Step 04 of 04

Like translations, rotations preserve size, shape, AND orientation (CCW vertex order stays CCW; rotations don't flip). Pre-image and image are congruent.

Key insight

Rotation = turn. Multiples of 90° about the origin have memorable rules. Direction matters: counter-clockwise positive, clockwise negative. Orientation is preserved (rotations don't flip).

8.F.4 — Sequences of transformations and congruence.

Step 01 of 04

You can apply MULTIPLE transformations in sequence. The result is itself a single rigid motion — congruent to the original.

Original translate Image 1 reflect Image 2

Step 02 of 04

Order matters. "Translate then reflect" usually doesn't equal "reflect then translate." Apply each transformation step-by-step.

A (1, 2) translate by ⟨ 3, 1 ⟩ (4, 3) reflect across x-axis (4, - 3)

Step 03 of 04

Two figures are CONGRUENT iff one can be mapped to the other by some sequence of translations, reflections, and rotations. This is the formal definition: rigid motions preserve congruence.

Step 04 of 04

Common shortcuts: two reflections across PARALLEL lines = translation. Two reflections across INTERSECTING lines = rotation about their intersection.

Key insight

Apply transformations one at a time, in the given order. Order generally matters. Two figures are congruent iff a sequence of rigid motions maps one to the other.

8.F.5 — Dilations and similarity.

Step 01 of 04

A dilation scales a figure UP or DOWN by a fixed factor about a center point. Unlike the three rigid motions, dilations CHANGE size — but they preserve shape (angles stay the same, sides scale proportionally).

Step 02 of 04

Dilation about the origin with scale factor $k$ :

(x, y) \to (k x, k y)

$k > 1$	→enlargement (image bigger)
$0 < k < 1$	→reduction (image smaller)
$k = 1$	→no change
$k < 0$	→enlarges/reduces AND rotates 180° (rare in basic problems)

Step 03 of 04

Worked example. Dilate $A (2, 3), B (4, 3), C (3, 6)$ by scale factor 2.

A^{'} = (4, 6), B^{'} = (8, 6), C^{'} = (6, 12)

Side lengths double; angles stay identical. Pre-image and image are SIMILAR.

Step 04 of 04

Two figures are similar iff a sequence of rigid motions AND dilations maps one to the other. Similar = same shape, possibly different size. The four transformations (translate, reflect, rotate, dilate) together describe everything geometric similarity allows.

Key insight

Dilation = scale by factor k. Multiplies coordinates (about the origin). Side lengths scale by k; angles preserved. Two figures are similar iff related by a sequence of rigid motions PLUS dilations.

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