A translation slides every point of a figure the same distance in the same direction. It is defined by a translation vector (a, b), which maps each point (x, y) to (x+a, y+b). Translations are rigid motions (isometries): they preserve distance, angle measure, and orientation. The image is congruent to the preimage. Translations can be described using vector notation or coordinate rules.
Start with sliding physical shapes on grid paper. Introduce vector notation and the coordinate rule. Practice translating individual points, then entire figures. Verify that distances and angles are preserved. Connect to real-world examples (sliding a puzzle piece, scrolling a screen).
You already know how to plot points and navigate the coordinate plane — now you can use that knowledge to move entire figures with mathematical precision. A translation is the simplest geometric transformation: every point in the figure slides the same distance in the same direction. If you translate a triangle 3 units right and 2 units up, every vertex moves exactly 3 right and 2 up. The shape does not rotate, flip, or resize — it simply relocates.
The coordinate rule captures this precisely. A translation by vector (a, b) maps every point (x, y) to (x + a, y + b). If a is positive, points move right; if negative, they move left. If b is positive, points move up; if negative, they move down. To translate a whole figure, apply the rule to each vertex and reconnect. For example, translating the point (2, 5) by vector (−3, 4) gives (2 + (−3), 5 + 4) = (−1, 9). The vector tells you both direction and distance in one compact notation.
Because every point moves by the same displacement, translations are isometries — they preserve distance, angle measure, and shape. The translated figure (called the image) is congruent to the original (the preimage). You can verify this with coordinates: if two points P and Q are distance d apart, their images P' and Q' are still distance d apart, because adding the same constant to both coordinates cancels out in the distance formula. Orientation is also preserved — clockwise stays clockwise — which distinguishes translations from reflections, which flip orientation.
Translations appear everywhere in mathematics and its applications. In computer graphics, sliding an object across the screen is a translation. In physics, a rigid body moving without rotating undergoes pure translation. In more advanced geometry, you will find that two translations compose to another translation (just add the vectors), but a translation followed by a rotation produces a more complex transformation. These composition rules make the set of all translations a group under composition — a structure that becomes important in abstract algebra. For now, the key intuition is that a translation is the purest kind of motion: everything moves together, nothing changes shape, and the rule is just addition.