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Location and Transformation
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New Zealand
Level 5

Translations on Cartesian Plane

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Practice
Lesson

Translations

A we saw in the previous lesson on Slides, A translation occurs when we move an object or shape from one place to another without changing its size, shape or orientation. Sometimes called a SLIDE, a translation moves every point on an object or shape the same distance in the same direction. We can translate points, lines or polygons on the Cartesian plane by moving them left, right, up or down any number of units.  

If we translate horizontally, only the $x$x value is changing.  

In this diagram, the image is a translation of $7$7units right. We would just write this as a translation of [7,0], indicating that it is $7$7 units, right (because it is positive) and no vertical movement because the y value is $0$0.

Notice how the coordinate of the vertex of the triangle has changed from $\left(-5,1\right)$(−5,1) to $\left(2,1\right)$(2,1) and that the $y$y coordinate has not changed at all.

 

If we translate vertically, only the y value is changing. 

In this diagram, the image is a translation of $5$5 units up.  We would just write this as a translation of [0,5], indicating that it is $5$5 units up (because it is positive) and no horizontal movement because the $x$x value is $0$0.

 

Notice how the coordinate of the vertex of the triangle has changed from $\left(1,-3\right)$(1,−3) to $\left(1,2\right)$(1,2) and that the $x$x coordinate has not changed at all.

 

As I just showed you, we can also indicate translations on the plane using special coordinate notation like this [x,y]

[1,0] is a translation of 1 unit right (positive) 

[0,-1] is a translation of 1 unit down (negative y)

and [-3,6] is a translation of 3 left AND 6 up.  

Have a quick play with this interactive to further consolidate the ideas behind translations on the Cartesian plane.  

Created with Geogebra

Let's have a look at these worked examples.

Question 1

What translation is required to get from triangle $ABC$ABC to triangle $A'B'C'$A′B′C′?

Loading Graph...

Two triangles are graphed in Cartesian coordinate plane marked from -10 to 10 in both $x$x- and $y$y- axes. In triangle $ABC$ABC, the coordinates of vertex $A$A is $\left(2,-10\right)$(2,−10), vertex $B$B is $\left(8,-10\right)$(8,−10), and vertex $C$C is $\left(2,-5\right)$(2,−5). While in triangle $A'B'C'$A′B′C′, the coordinates of vertex $A'$A′ is $\left(-8,-10\right)$(−8,−10), vertex $B'$B′ is $\left(-2,-10\right)$(−2,−10), and vertex $C'$C′ is $\left(-8,-5\right)$(−8,−5). 

  1. $9$9 units to the left

    A

    $10$10 units to the left

    B

    $9$9 units to the right

    C

    $10$10 units to the right

    D

Question 2

What is the translation of the trapezium $ABCD$ABCD to the trapezium $A'B'C'D'$A′B′C′D′?

Loading Graph...

A Cartesian plane is presented with x- and y-axes ranging from -10 to 10. On the plane, there are two shaded trapeziums. One trapezium has vertices labeled $A$A, $B$B, $C$C, and $D$D with vertex $A$A at A(-1, -1) and vertex $D$D at D(4, -1) forming the longer base. Vertex $B$B at B(0, -9) and vertex $C$C at C(3, -9) form the shorter base. 

The other trapezium is labeled $A'$A′, $B'$B′, $C'$C′, and $D'$D′, with similar positioning of the vertices relative to the trapezium on the left side where vertex  $A'$A′ is at A'(-10, 2), vertex $B'$B′is at B'(-9, -6), vertex $C'$C′is at C'(-6, -6) and vertex $D'$D′is at D'(-5, 2).

  1. $9$9 units left and $3$3 units up

    A

    $3$3 units right and $9$9 units up

    B

    $3$3 units left and $9$9 units down

    C

    $9$9 units right and $3$3 units down

    D

Question 3

Consider the translation from each shape labelled $A$A to the same shape labelled $B$B. Which of these shapes is translated according to [$-11$−11, $-8$−8]?

(For example, the notation [$1$1, $-2$−2] represents a horizontal translation $1$1 unit to the right and a vertical translation $2$2 units down.)

Loading Graph...

  1. square

    A

    triangle

    B

    rectangle

    C

Question 4

The point undergoes a translation of [$4$4, $-2$−2]. Place the new point on the graph resulting from this translation.

(The notation [$1$1, $-2$−2] represents a horizontal translation $1$1 unit to the right and a vertical translation $2$2 units down.)

  1. Loading Graph...

Outcomes

GM5-9

Define and use transformations and describe the invariant properties of figures and objects under these transformations

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