NZ Level 5

Translations on Cartesian Plane

Lesson

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.

Let's have a look at these worked examples.

What is the translation of triangle $ABC$`A``B``C` to triangle $A'B'C'$`A`′`B`′`C`′?

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$11$11 units right

A$10$10 units right

B$11$11 units left

C$10$10 units left

D$11$11 units right

A$10$10 units right

B$11$11 units left

C$10$10 units left

D

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

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$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$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

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.)

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square

Atriangle

Brectangle

Csquare

Atriangle

Brectangle

C

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.)

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