Skill: Translate figures on a coordinate plane

A translation slides every point of a figure the same distance and in the same direction.

A translation vector $\langle a,b\rangle$ tells you:

• $a$: move horizontally—right if positive, left if negative
• $b$: move vertically—up if positive, down if negative

Use the coordinate rule

$$(x,y)\rightarrow(x+a,y+b).$$

Example

Translate triangle $ABC$ with vertices

$$A(-2,1),\quad B(1,1),\quad C(-2,4)$$

by the vector $\langle 3,-2\rangle$.

The $3$ means move $3$ units right. The $-2$ means move $2$ units down. Add $3$ to every $x$-coordinate and subtract $2$ from every $y$-coordinate:

$$(x,y)\rightarrow(x+3,y-2).$$

Apply the rule to each vertex:

For $A(-2,1)$:

$$A'(-2+3,1-2)=(1,-1)$$

For $B(1,1)$:

$$B'(1+3,1-2)=(4,-1)$$

For $C(-2,4)$:

$$C'(-2+3,4-2)=(1,2)$$

So the translated triangle has vertices

$$\boxed{A'(1,-1),\quad B'(4,-1),\quad C'(1,2)}.$$

Be sure to apply the same vector to every point. A translation keeps the figure’s side lengths, angle measures, shape, and orientation unchanged; it only changes the figure’s location.