Translate figures on a coordinate plane

A translation moves every point of a figure the same signed horizontal and vertical distance, represented by a translation vector ⟨a, b⟩ and the coordinate rule $(x,y)\rightarrow(x+a,y+b)$ ; positive and negative components distinguish right/left and up/down shifts. The learner understands that translations preserve lengths, angle measures, orientation, and congruence, and avoids reversing signs or shifting only selected points. The scope centers on coordinate-plane translations, not matrix methods or generalized compositions of transformations.

Detailed Explanation: 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.

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Cartesian Grid - Translation of Shape in Two Dimensions (Words)

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