Consider the following coordinate plane.

When ΔABC is transformed to ΔA′B′C′, AB and A′B′ are parallel to the x-axis, and AC and A′C′ are parallel to the y-axis. Moreover, ΔABC and ΔA′B′C′ have the same orientation. i.e., the way they face is the same. This type of transformation is said to be a translation.

If point P is translated to point P′, then the vector

is said to be the translation vector.

If u = (h, k) is a translation vector, then the image of the point (x, y) under the translation will be the point (x + h, y + k).

The above example shows that a translation is a rigid motion, a motion that does not change shape and size of figures.

Translation formula in terms of coordinates of the plane.

1. If (h, k) is a translation vector, then
(a) the origin is translated to (h, k) i.e., (0, 0)→ (h, k)
(b) the point P (x, y) is translated to P′ (x + h, y + k).

2. If the translation vector is
 where A = (a, b) and B = (c, d), then
(a) the origin is translated to (c – a, d – b), and
(b) the point P(x, y) is translated to (x + c – a, y + d – b)