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.
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.