Translations

A translation is a transformation that slides every point of a figure the same distance in the same direction. Nothing rotates and nothing flips — the figure simply glides to a new position. A translation is a , so the image stays congruent to the original.

Translating by a vector

The slide is described by a translation vector $\langle a, b \rangle$ , where $a$ is the horizontal shift and $b$ is the vertical shift.

Make a prediction: The point (2, 3) is translated by the vector ⟨5, −1⟩ — 5 units right and 1 unit down. Where does its image land?

(7, 2)(7, 4)(−3, 4)(10, −3)

Applying the vector to a point adds its components coordinate-wise:

(x, y) \mapsto (x + a,\, y + b)

For example, translating the point $(1, 1)$ by $\langle 3, -2 \rangle$ moves it to $(4, -1)$ . Because the same vector is applied to every vertex, the shape keeps its size, its orientation, and the order of its vertices — only its location changes.

Drag the shift sliders and watch triangle ABC slide to A′B′C′.

Shift x3

Shift y-2

A triangle ABC translated by the vector (3, -2) → triangle A′B′C′.

Practice

Score: 0/2

1.Translate the point (2, 5) by the vector ⟨4, -3⟩. Where does it land?
(6, 2)(-2, 8)(6, 8)(2, 2)
2.A vertex sits at (−1, 4). After translating by ⟨6, 2⟩, what is its new y-coordinate?
Answer

For teachers

Standard, common misconceptions, and suggested use will live here.