Skill: Perform translations

A translation slides a figure without turning or flipping it. Every point moves the same number of units in the same direction.

If the translation moves a point:

• $a$ units horizontally, and
• $b$ units vertically,

then use the rule

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

Add $a$ to every $x$-coordinate and add $b$ to every $y$-coordinate.

Example

Translate triangle $ABC$ with vertices

$$A(1,2),\quad B(4,2),\quad C(2,5)$$

$3$ units right and $2$ units down.

Moving $3$ units right means add $3$ to each $x$-coordinate. Moving $2$ units down means subtract $2$ from each $y$-coordinate. The rule is

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

Apply the rule to each vertex:

For $A(1,2)$:

$$A'(1+3,\ 2-2)=A'(4,0)$$

For $B(4,2)$:

$$B'(4+3,\ 2-2)=B'(7,0)$$

For $C(2,5)$:

$$C'(2+3,\ 5-2)=C'(5,3)$$

The translated triangle has vertices

$$A'(4,0),\quad B'(7,0),\quad C'(5,3).$$

Connect the new points in the same order. The new triangle has the same side lengths, angles, shape, and orientation as the original; it has simply been slid $3$ units right and $2$ units down.