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# How to Translate a Point

#### What is a Translation?

Diagram 1

The animation below shows a triangle being translated.

### 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

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

### 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

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

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

More like Example 1...

### Practie Problems

##### Problem 1

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

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

##### Problem 2

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

$(-4, 3 ) \text{ under } T_{(\red{-7}, \red{-3 })} \\ (-4 \red{-7}, 3 \red{-3}) \\ \boxed{(-11, 0 )}$
What is the image of point B , $$(-2a, 4b)$$ under translation of $$T_{(+3a, +5b )}$$ ?
$(-2a, 4b ) \text{ under } T_{(\red{+3a }, \red{ +5b })} \\ (-2a \red{+3a}, 4b \red{+5b}) \\ \boxed{(a, 9b )}$