Perform translations

A translation moves every point of a figure the same distance in the same direction, preserving lengths, angle measures, shape, and orientation; corresponding points therefore have equal horizontal and vertical changes. On a coordinate plane, the image of $(x,y)$ under a translation by $(a,b)$ is $(x+a,y+b)$ , with coordinates and shifts represented using integers; rotations, reflections, and formal vector methods are not included.

Detailed Explanation: 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.

Learn by doing: Perform translations

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Cartesian Grid - Translation of Point (Coordinates to Coordinates) in Two Dimensions (Vector)

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