Translating Points and Figures

A translation is a slide: pick up a figure and set it down somewhere else without turning it, flipping it, or resizing it. Because every single point moves by the exact same horizontal amount and the exact same vertical amount, the rule is simple to write: (x, y) becomes (x + h, y + k). Moving right or left only changes x, since left-right motion is horizontal. Moving up or down only changes y, since that motion is vertical. The two directions never interfere with each other, which is why sliding a triangle 3 units right adds 3 to every x-coordinate and leaves every y-coordinate exactly as it was.

To translate a whole figure, apply the same rule to each vertex separately, then reconnect them in the same order. Take the triangle with vertices (1, 2), (4, 2), (4, 6), translated left 5 and up 3: subtract 5 from each x-coordinate and add 3 to each y-coordinate. The vertices become (-4, 5), (-1, 5), (-1, 9). The triangle is in a new spot, but it is the same size and shape, just shifted.

Because a translation is a rigid motion, it preserves every distance and every angle, which means it also preserves slope: a side's direction never changes, only its location. This is what separates translations from the other transformations in this unit. Reflections flip direction, rotations turn it, and dilations resize — translations alone leave slope, angle measure, and side length completely untouched, which makes them the simplest rigid motion to check on a figure.