Unit 3 Part 1: Translation

Translations

"Slide to the left
Slide to the right
Take it back now y'all"
- Euclid of Alexandria, OG of Geometry

Graphing

Warm Up: Plot the following points

(-2, -2), (-1, 1), (1, 0), (2, -2), (2, -3),
(-1, -2), (1, -3), (0, -2), (1, -2)

​Translation

• Translations change the position of a figure

• They slide a shape in any direction

Translation Rules

General Rules for Translations

​Coordinate Rule	Direction​
x + #​	right​
x - #​	left​
y + #​	up​
y - #​	down​

Translate ΔABC by (x - 4 , y + 5)

• ​ΔABC is the pre-image (blue Δ)

  • (Triangle "A B C")

• ΔA'B'C' is the post-image (red Δ)

  • (Triangle "A B C" prime)

Translate ΔABC by (x - 4 , y + 5)

• The rule for this translation is
  (x - 4 , y + 5)

  • Meaning the blue triangle moves left 4 places and up 5 places

  • Mathematically, we will subtract 4 from the x coordinates and add 5 to the y coordinates

Translate ΔABC by (x - 4 , y + 5)

• In order to translate the entire shape, you must translate each individual point

• Coordinates for ΔABC

  • Point A lies at ( 3 , -1)

  • Point B lies at ( 7 , -5)

  • Point C lies at (-2 , -2)

• To translate each individual point subtract 4 from each x coordinate and add 5 to each y coordinate

Translate ΔABC by (x - 4 , y + 5)

​X Coordinates	​Point A	​Point B	​Point C
​Old X Coordinate	3​	​	​
​Rule: x - 4	​	​	​
​New X Coordinate	​	​	​

​Y Coordinates	​Point A	​Point B	​Point C
​Old Y Coordinate	-1	​	​
​Rule: y + 5	​	​	​
​New Y Coordinate	​	​	​

• Coordinates for ΔABC translated to ΔA'B'C'

  • Point A ( 3 , -1)  becomes...

  • Point B ( 7 , -5) becomes...

  • Point C (-2 , -2) becomes...

Translate ΔABC by (x - 4 , y + 5)

​X Coordinates	​Point A	​Point B	​Point C
​Old X Coordinate	3​	​	​
​Rule: x - 4	3 - 4​	​	​
​New X Coordinate	​	​	​

​Y Coordinates	​Point A	​Point B	​Point C
​Old Y Coordinate	-1	​	​
​Rule: y + 5	-1 + 5	​	​
​New Y Coordinate	​	​	​

• Coordinates for ΔABC translated to ΔA'B'C'

  • Point A ( 3 , -1)  becomes...

  • Point B ( 7 , -5) becomes...

  • Point C (-2 , -2) becomes...

Translate ΔABC by (x - 4 , y + 5)

​X Coordinates	​Point A	​Point B	​Point C
​Old X Coordinate	3​	​	​
​Rule: x - 4	3 - 4​	​	​
​New X Coordinate	-1​	​	​

​Y Coordinates	​Point A	​Point B	​Point C
​Old Y Coordinate	-1	​	​
​Rule: y + 5	-1 + 5	​	​
​New Y Coordinate	4	​	​

• Coordinates for ΔABC translated to ΔA'B'C'

  • Point A ( 3 , -1)  becomes A' (-1 , 4)

  • Point B ( 7 , -5) becomes...

  • Point C (-2 , -2) becomes...

Translate ΔABC by (x - 4 , y + 5)

​X Coordinates	​Point A	​Point B	​Point C
​Old X Coordinate	3​	7​
​Rule: x - 4	3 - 4​
​New X Coordinate	-1​

​Y Coordinates	​Point A	​Point B	​Point C
​Old Y Coordinate	-1	-5
​Rule: y + 5	-1 + 5
​New Y Coordinate	4

• Coordinates for ΔABC translated to ΔA'B'C'

  • Point A ( 3 , -1)  becomes A' (-1 , 4)

  • Point B ( 7 , -5) becomes...

  • Point C (-2 , -2) becomes...

Translate ΔABC by (x - 4 , y + 5)

​X Coordinates	​Point A	​Point B	​Point C
​Old X Coordinate	3​	7​	​
​Rule: x - 4	3 - 4​	7 - 4​	​
​New X Coordinate	-1​	​	​

​Y Coordinates	​Point A	​Point B	​Point C
​Old Y Coordinate	-1	-5	​
​Rule: y + 5	-1 + 5	-5 + 5	​
​New Y Coordinate	4	​	​

• Coordinates for ΔABC translated to ΔA'B'C'

  • Point A ( 3 , -1)  becomes A' (-1 , 4)

  • Point B ( 7 , -5) becomes...

  • Point C (-2 , -2) becomes...

Translate ΔABC by (x - 4 , y + 5)

​X Coordinates	​Point A	​Point B	​Point C
​Old X Coordinate	3​	7​	​
​Rule: x - 4	3 - 4​	7 - 4​	​
​New X Coordinate	-1​	3​	​

​Y Coordinates	​Point A	​Point B	​Point C
​Old Y Coordinate	-1	-5	​
​Rule: y + 5	-1 + 5	-5 + 5	​
​New Y Coordinate	4	0	​

• Coordinates for ΔABC translated to ΔA'B'C'

  • Point A ( 3 , -1)  becomes A' (-1 , 4)

  • Point B ( 7 , -5) becomes B' (3 , 0)

  • Point C (-2 , -2) becomes...

Translate ΔABC by (x - 4 , y + 5)

​X Coordinates	​Point A	​Point B	​Point C
​Old X Coordinate	3​	7​	​-2
​Rule: x - 4	3 - 4​	7 - 4​	​
​New X Coordinate	-1​	3​	​

​Y Coordinates	​Point A	​Point B	​Point C
​Old Y Coordinate	-1	-5	​-2
​Rule: y + 5	-1 + 5	-5 + 5	​
​New Y Coordinate	4	0	​

• Coordinates for ΔABC translated to ΔA'B'C'

  • Point A ( 3 , -1)  becomes A' (-1 , 4)

  • Point B ( 7 , -5) becomes B' (3 , 0)

  • Point C (-2 , -2) becomes...

Translate ΔABC by (x - 4 , y + 5)

​X Coordinates	​Point A	​Point B	​Point C
​Old X Coordinate	3​	7​	​-2
​Rule: x - 4	3 - 4​	7 - 4​	​-2 - 4
​New X Coordinate	-1​	3​	​

​Y Coordinates	​Point A	​Point B	​Point C
​Old Y Coordinate	-1	-5	​-2
​Rule: y + 5	-1 + 5	-5 + 5	​-2 + 5
​New Y Coordinate	4	0	​

• Coordinates for ΔABC translated to ΔA'B'C'

  • Point A ( 3 , -1)  becomes A' (-1 , 4)

  • Point B ( 7 , -5) becomes B' (3 , 0)

  • Point C (-2 , -2) becomes...

Translate ΔABC by (x - 4 , y + 5)

​X Coordinates	​Point A	​Point B	​Point C
​Old X Coordinate	3​	7​	​-2
​Rule: x - 4	3 - 4​	7 - 4​	​-2 - 4
​New X Coordinate	-1​	3​	​-6

​Y Coordinates	​Point A	​Point B	​Point C
​Old Y Coordinate	-1	-5	​-2
​Rule: y + 5	-1 + 5	-5 + 5	​-2 + 5
​New Y Coordinate	4	0	​3

• Coordinates for ΔABC translated to ΔA'B'C'

  • Point A ( 3 , -1)  becomes A' (-1 , 4)

  • Point B ( 7 , -5) becomes B' (3 , 0)

  • Point C (-2 , -2) becomes C' (-6 , 3)

Translate ΔABC by (x - 4 , y + 5)

• Thus, we get the image to the right

Translate ΔABC by (x - 4 , y + 5)

• Thus, we get the image to the right

Drag and Drop

Coordinates for ΔABC translated to ΔA'B'C':

Rule: ( x + 2 , y - 6 )

Point A ( 3 , -1 ) becomes A' (

Placeholder

,

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)
Point B ( 7 , -5 ) becomes B' (

Placeholder

,

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)
Point C ( -2 , -2 ) becomes C' (

Placeholder

, -8)

Drag these tiles and drop them in the correct blank above

5

-7

-11

0

9

-5

10

-3

Drag and Drop

Coordinates for ΔBGT translated to ΔB'G'T':

Rule: ( x + 5 , y + 1 )

Point B ( -3 , -5 ) becomes B' (

Placeholder

,

Placeholder

)
Point G ( -3 , -1 ) becomes G' (

Placeholder

, 0 )
Point T ( -1 , -1 ) becomes T' (

Placeholder

,

Placeholder

)

Drag these tiles and drop them in the correct blank above

2

-4

0

4

-8

1

-3

-2

Drag and Drop

Coordinates for ΔYGM translated to ΔY'G'M':

Rule: ( x - 1 , y + 2 )

Point A ( -3 , 0 ) becomes A' (

Placeholder

, 2 )
Point B ( 0 , -2 ) becomes B' (

Placeholder

,

Placeholder

)
Point C ( -3 , -4 ) becomes C' (

Placeholder

,

Placeholder

)

Drag these tiles and drop them in the correct blank above

-2

-4

0

-1

-3

1

5

-5