02.01 Translations and Reflections

Two rigid moves

A translation slides a figure. No turn. No flip. Every point travels the same distance in the same direction.

A reflection flips a figure across a line. The image is a mirror. Size and shape stay. Orientation (clockwise vs counterclockwise naming) reverses.

Both are rigid motions, so the image is congruent to the pre-image. Collinearity and betweenness stay: if D is the midpoint of AB, then D' is the midpoint of A'B'.

Lookalike: a translation keeps orientation. A reflection does not. Congruence does not care. If a question asks "are they congruent," the answer is yes for both. If it asks "did the naming direction reverse," only the reflection did.

The slide rule

Count right/left first (x), then up/down (y). Right and up are positive. Left and down are negative.

ΔABC with A(3, 6), B(1, 4), C(7, 4) moves 1 right and 3 down. A' is (4, 3). Same count from B and from C. One rule for the whole triangle:

(x, y) → (x + 1, y − 3)

Blue is the pre-image. Red is the image. Each vertex used the same (x + 1, y − 3).

If even one vertex used a different count, the triangles would not be congruent. Same rule, every point.

Two ways to get the image

Hexagon WATERS with W(3, 5). Rule: (x, y) → (x − 4, y − 7).

Method one: plug the numbers. W(3, 5) → (3 − 4, 5 − 7) = W'(−1, −2).

Method two: stand on W and count 4 left, 7 down.

Same two methods work in reverse. Given a pre-image and an image, pick any matched pair, count x then y, write the rule, then check a second pair so you did not mix vertices.

Same pair, two reasons: a quiz can show W' at (−1, −2) in two choices. One names the translation rule. One names "because −4 and −7" as if that were a Property of Equality. The move is a mapping of points, not solving an equation. Pick the translation rule.

Vectors, quickly

A vector has length and direction. Initial point is the tail. Terminal point is the arrowhead. Unlike a ray, a vector has a fixed length.

When you draw the vector from each pre-image vertex to its image, a translation makes a set of parallel, congruent arrows. That is the picture of "every point moved the same way."

The frog at (0, 0) that hops 3 right and 2 up is the rule (x + 3, y + 2). Input (0, 0), output (3, 2).

The three reflection rules

The line of reflection is the perpendicular bisector of every segment that joins a point to its image. Equidistant, and perpendicular.

On the coordinate plane, Canvas wants these three as functions:

Across the x-axis (the line y = 0): (x, y) → (x, −y)

Across the y-axis (the line x = 0): (x, y) → (−x, y)

Across the line y = x: (x, y) → (y, x)

P(5, −3) as a check:

x-axis → (5, 3)
y-axis → (−5, −3)
y = x → (−3, 5)

Lookalike rules: (y, x) swaps. (x, −y) flips the y-sign. (y, −x) is 90° clockwise, which is a rotation, not a reflection over y = x. If two choices share the digits 3 and 5, read the axis name in the stem before you pick.

Sequences

Canvas stacks moves. Do them in the order given. Each output is the next input.

Start at (−5, 2).

Reflect over the x-axis → (−5, −2)
Reflect over the y-axis → (5, −2)
Reflect over y = x → (−2, 5)

Two reflections in a row can look like a translation or a rotation of the original. That is allowed. Name each step by the rule in that step, not by what the whole chain "sort of looks like" at the end, unless the question asks for a single equivalent move.

Your turn

New figures, same rules. Watch for two choices that share an ordered pair.