Skill: Translate figures on a coordinate plane

A translation slides a figure without turning it, flipping it, or changing its size.

The rule

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

means:

• Add $a$ to every $x$-coordinate. This moves the figure horizontally.
• Add $b$ to every $y$-coordinate. This moves the figure vertically.
• A positive number moves right or up.
• A negative number moves left or down.

Example

Translate triangle $ABC$ with vertices

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

using the rule

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

Step 1: Identify the coordinate changes

The rule says:

• $x-3$: move every point $3$ units left.
• $y+2$: move every point $2$ units up.

Step 2: Apply the rule to each vertex

For $A(1,2)$:

$$(1,2)\rightarrow(1-3,2+2)=(-2,4)$$

So, $A'=(-2,4)$.

For $B(4,2)$:

$$(4,2)\rightarrow(4-3,2+2)=(1,4)$$

So, $B'=(1,4)$.

For $C(2,5)$:

$$(2,5)\rightarrow(2-3,5+2)=(-1,7)$$

So, $C'=(-1,7)$.

Step 3: State the translated figure

The translated triangle has vertices

$$A'(-2,4),\quad B'(1,4),\quad C'(-1,7).$$

Every vertex moved $3$ units left and $2$ units up. The triangle keeps the same side lengths, angle measures, orientation, and shape; only its location changes.