Debug
animation of translation

How to Translate a Point

What is a Translation?

Answer:

It's all about sliding

Diagram 1

The animation below shows a triangle being translated.

animation of a translation in math

Translations are Isometries

Translations are isometries . As you can see in diagram 2 below, $$ \triangle ABC $$ is translated to form its image $$ \triangle A'B'C' $$. And the distance between each of the points on the preimage is maintained in its image

Diagram 2

translations are isometries

The length of each segment of the preimage is equal to its corresponding side in the image .

$ m \overline{AB} = 3 \\ m \overline{A'B'} = 3 \\ \\ m \overline{BC} = 4 \\ m \overline{B'C'} = 4 \\ \\ m \overline{CA} = 5 \\ m \overline{C'A'} = 5 $


You can also see that the orientation of the letters is preserved, i.e. that the vertices in the original shape are ordered $$ABC$$ , in clockwise order, and that the image maintains the same clockwise order, making translations a direct isometry.

Diagram 3
translations preserve orientation

Translation Notation

In translation notation, the first number represents how many units in the x direction, the second number, how many in the y direction. Both numbers tell us about how far and in what direction we are going to slide the point.

Diagram 4
translations notation

In the animation below, you can see how we actually translate the point by $$-1 $$ in the x direction and then by $$ +2$$ in the y direction .

Diagram 5
how to translate point animation

As the animation shows a translation of $$T_{(\red{-1,+2})} $$ on the point A with coordinates $$(3,2) $$ produces an image at $$ (2,4) $$.

And $$ (3 \red{-1}, 2 \red{+2}) \rightarrow (2,4) $$.

Examples of Translating points

Example 1
Picture of a translation in math

More like Example 1...

Picture of a translation in math
Graph of a translation in math

Practie Problems

Problem 1

What is the image of point B , $$ (1,5) $$ under translation of $$ T_{(-3,+2)} $$ ?

Follow the examples above and add the x,y coordinates :

$ (1, 5) \text{ under } T_{(\red{-3}, \red{+2})} \\ (1 \red{-3}, 5 \red{+2}) \\ \boxed{(-2, 7)} $

Problem 2

What is the image of point B , $$ (-4,3) $$ under translation of $$ T_{(-7,-3 )} $$ ?

Follow the examples above and add the x,y coordinates :

$ (-4, 3 ) \text{ under } T_{(\red{-7}, \red{-3 })} \\ (-4 \red{-7}, 3 \red{-3}) \\ \boxed{(-11, 0 )} $

Problem 3

What is the image of point B , $$ (-2a, 4b) $$ under translation of $$ T_{(+3a, +5b )} $$ ?

Follow the examples above and add the x,y coordinates :

$ (-2a, 4b ) \text{ under } T_{(\red{+3a }, \red{ +5b })} \\ (-2a \red{+3a}, 4b \red{+5b}) \\ \boxed{(a, 9b )} $


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