A translation is a transformation that slides a figure across the plane without turning it, flipping it, stretching it, or shrinking it. Every point of the figure moves the same distance in the same direction. Translations are important because they show how location can change while size, shape, and orientation stay the same.
They are used in geometry, computer graphics, map movements, and animation.
Understanding Geometry: Translations on the Plane
A translation can be described by one displacement vector. A vector is an instruction with two parts, one horizontal change and one vertical change. It has no fixed starting point.
This matters because the same vector can be placed beside any vertex and still show the correct movement. On graph paper, drawing an arrow from an original point to its image helps make the instruction visible.
Arrows for several corresponding points should be parallel, point the same way, and have equal length. If one arrow differs, the mapping is not a translation.
On a coordinate grid, the horizontal coordinate changes by a constant amount for every point. The vertical coordinate changes by another constant amount for every point. Adding a positive amount shifts a coordinate in the positive direction, while adding a negative amount shifts it in the negative direction.
A useful method is to make a table of original vertices and image vertices. Compare each pair by finding the horizontal difference and the vertical difference.
The two differences must match for every row. This method is more reliable than judging a diagram by eye, especially when a figure has many vertices or the grid is not drawn clearly.
Translations are useful in proofs because they preserve the measurements that define a shape. A triangle and its translated image have matching side lengths and matching angle measures. Corresponding segments remain parallel because their direction has not changed.
This lets students show that the two figures are congruent by tracking each vertex. A translation can be combined with another translation. The overall result is found by combining the horizontal changes and combining the vertical changes.
A reverse translation undoes the first one. For example, a shift right by four units is undone by a shift left by four units, with the same idea applying vertically.
Students meet translations when moving a game character across a screen, panning a digital map, arranging repeated tiles, or placing a copied object in a drawing program. In computer graphics, each point of an object is updated by the same displacement so the object keeps its position relative to itself. Common errors come from mixing up horizontal and vertical changes, using the wrong sign, or moving only one vertex correctly.
Another mistake is to redraw the figure from memory instead of mapping each labeled point. Label original points and image points carefully, then connect corresponding pairs mentally or with light arrows. Checking one side length after the move can catch a plotting error, but checking the shared displacement for every vertex is the strongest test.
Key Facts
- A translation moves every point the same distance in the same direction.
- Coordinate rule: (x, y) -> (x + a, y + b).
- Vector notation: <a, b> means move a units horizontally and b units vertically.
- Positive a moves points right, and negative a moves points left.
- Positive b moves points up, and negative b moves points down.
- Translations preserve distance, angle measure, parallel lines, size, shape, and orientation.
Vocabulary
- Translation
- A transformation that slides every point of a figure the same distance in the same direction.
- Vector
- A quantity with both direction and distance that describes how far and where a figure moves.
- Image
- The new figure formed after a transformation is applied.
- Preimage
- The original figure before a transformation is applied.
- Coordinate Rule
- A rule that shows how each point's coordinates change during a transformation.
Common Mistakes to Avoid
- Adding the horizontal change to the y-coordinate is wrong because left and right movement changes x, not y.
- Forgetting negative signs in the vector is wrong because <3, -2> moves right 3 and down 2, not right 3 and up 2.
- Moving only one vertex of a polygon is wrong because a translation requires every point of the figure to move by the same vector.
- Changing the size or rotating the figure is wrong because a translation preserves size, shape, and orientation.
Practice Questions
- 1 Translate point A(2, -5) by the vector <4, 3>. What are the coordinates of A'?
- 2 Triangle PQR has vertices P(-1, 2), Q(3, 2), and R(1, 5). Translate the triangle using the rule (x, y) -> (x - 4, y + 1). Find P', Q', and R'.
- 3 A figure is translated by <0, -6>. Explain what happens to the figure's position, size, shape, and orientation.