5 right, 2 up — every point makes the same move. The blue flag is the original; the green flag is its image; each amber arrow records one point's trip, and the arrows are all the same length and all parallel.
A sliding door, a box on a conveyor belt, an elevator climbing its shaft: the whole thing moves as one. Every part travels the same way by the same amount, and nothing turns or flips along the way. That motion is a translation — a clean slide. And here is the surprise that closes out our stage: the little arrows that record each point's trip turn out to be exactly the equal, parallel segments you have been studying. So translation is parallel lines set in motion.
14.6.1 What translation is
A translation slides every point of a figure the same distance in the same direction. There is no turning (that would be a rotation) and no flipping (that would be a reflection) — just one honest slide. Think of pushing a book straight across a desk: it ends up somewhere new, but it faces the same way and looks exactly the same.
We give the two figures names. The figure you start with is the original (also called the pre-image); the figure you land on is the image. By custom we tag the image's points with a little prime mark: point A slides to A′ (read "A-prime"), B to B′, and so on.
One slide, three ways. Translation keeps the figure facing the same way; a rotation turns it; a reflection flips it to a mirror image. Only the first is a translation.
Key idea
A translation is a pure slide: same distance, same direction, for every point. No turn. No flip.
14.6.2 Describing a translation on a grid
Saying "slide it that way a bit" is not math. On grid paper we pin a translation down with exactly two counts: how many squares right, and how many squares up. Together those two numbers are the translation's vector — for example "5 right, 2 up."
The two directions that feel like "backwards" simply get the negative sign. Sliding left is negative-right, and sliding down is negative-up. So "3 left, 1 down" is the vector −3 right, −1 up. Choosing right and up as the two positive directions matches the coordinate grid you already know, where moving right grows the x-count and moving up grows the y-count.
Example
The point (2, 1) under the vector "3 right, 2 up" lands at (2 + 3, 1 + 2) = (5, 3). You add the right-count to x and the up-count to y. That is all a translation does to a point.
Try it Set the slide vector
Dial in the squares right and the squares up. Watch the green image move and the amber arrows stay equal and parallel.
squares right4
squares up2
14.6.3 The properties of translation
What does sliding do to a shape, and what does it leave alone? Almost everything is left alone — that is what makes a translation so well-behaved. Three properties capture it:
First, the image is congruent to the original: same size, same shape. Every length is unchanged and every angle is unchanged. We write this with the congruence symbol ≅, so for a triangle we would say △ABC ≅ △A′B′C′. Second, every point travels the same distance in the same direction — that is the very definition of the slide. Third, matching segments come out parallel and equal: side AB and its image A′B′ are the same length and point the same way.
Nothing about the shape changes. Matching sides carry the same tick marks (equal length) and matching angles carry the same arc (equal angle), so △ABC ≅ △A′B′C′.
Three properties of a translation
1. The image is congruent to the original (≅) — same lengths, same angles.
2. Every point moves the same distance in the same direction.
3. Each segment and its image are parallel and equal.
14.6.4 Drawing the image
Drawing a translated figure is wonderfully mechanical. You never have to redraw the shape by eye. Instead, follow the recipe one corner at a time:
Step 1. Take each vertex (corner) of the figure. Step 2. Move it by the vector — "so many right, so many up." Step 3. Once all the corners have moved, reconnect them in the same order. Because every corner shifted by the same amount, the reconnected shape is an exact copy, simply slid over.
Try it Build the image, one corner at a time
The vector is fixed at 5 right, 2 up. Step up to move the next corner; reach the last corner and the green image closes.
corners moved0
Watch out
Move every corner by the same vector. Slip on even one corner — moving it 4 right instead of 5 — and the copy is warped: lengths change, angles change, and it is no longer a translation at all.
14.6.5 Translation is parallel lines in action
Now connect the dots — literally. Draw an arrow from each original point to its image: from A to A′, from B to B′, and so on. Because every point made the same trip, these connector arrows are all the same length and all point the same way. A bundle of equal, same-direction segments is precisely a set of parallel and equal segments — exactly the idea you proved in 14.4 and 14.5.
The bundle of connectors. Lines AA′, BB′, CC′ are equal in length and parallel — parallel lines, in motion.
So our stage ends where it began. Crossing lines gave us families of angles; a transversal named three pairs; equal pairs proved lines parallel, and parallel lines handed those pairs back as properties. Translation now takes that parallel-and-equal idea and sets it moving across the page. It is also your first taste of congruence and transformations, the heart of the geometry to come in Stage 15 and beyond.
★ Recap
A translation slides every point of a figure the same distance in the same direction — no turn, no flip. On a grid you describe it by a vector: so many squares right and so many up (left and down being the negatives). To translate a point, add the right-count to x and the up-count to y, so (2, 1) under "3 right, 2 up" becomes (5, 3).
The image is congruent (≅) to the original: same lengths, same angles, same shape. To draw it, move every vertex by the same vector and reconnect in order. The connectors joining each point to its image are all parallel and all equal — translation is parallel lines in action.
✎ Exercises
Translate the point (2, 1) by the vector "3 right, 2 up." Where does it land?
Answer
Add the counts: (2 + 3, 1 + 2) = (5, 3).
A translation takes A(1, 1) to A′(6, 3). Describe the slide as a vector.
Answer
x went up by 6 − 1 = 5 and y went up by 3 − 1 = 2, so the vector is "5 right, 2 up."
Under a translation, is the image bigger, smaller, or the same size as the original? Why?
Answer
The same size — a translation is a rigid slide, so the image is congruent (≅) to the original: lengths and angles are unchanged.
A segment and its image after a translation — what two things are always true about them?
Answer
They are parallel and equal in length. Every point moved the same way by the same amount, so the matching segment is just a copy slid over.
Translate the triangle with vertices A(1, 2), B(4, 2), C(1, 5) by the vector "2 right, 1 up." List the image vertices.
Answer
Add (2, 1) to each: A′(3, 3), B′(6, 3), C′(3, 6).
True or false: a translation can turn a figure so it faces a new direction.
Answer
False. Turning is a rotation, not a translation. A translation slides without any turning — the figure keeps facing the same way.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson treats translation as a rigid motion: every point of the figure moves the same distance in the same direction, so the image is congruent to the original. On a grid we describe the slide with a vector — a "right, up" pair — and apply it to a point by adding the counts to its coordinates. Students draw the image by translating each vertex and reconnecting, then see that the connector segments joining matching points form a bundle of equal, parallel arrows. That observation deliberately ties the whole stage together: translation is the parallel-and-equal relationship from 14.4–14.5 set in motion.
The misconception to watch is confusing a slide with a turn or a flip — calling a rotation or a reflection a "translation." A close cousin is moving the corners by different amounts, which warps the figure and breaks congruence. Insist on the same vector for every vertex, and have students check that each connector arrow comes out the same length and parallel to the others.
This supports CCSS 8.G.A.1 (the properties of translations — lines map to lines, segments to equal segments, parallels are preserved), 8.G.A.3 (describing translations using coordinates), and HS G-CO.A.4 / G-CO.B.6 (translation as a rigid motion and its role in defining congruence).
eastmath.com · Stage 14 · 14.6 Translating a Figure · Reasoning, one step at a time
eastmath.com · 14.6 Translating a Figure · 14.6.3 The properties of translation