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9. Transformations & Congruence
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Math I

9.01 Translations

Lesson
Practice
Lesson

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 $xy$xy-plane by moving them left, right, up or down any number of units.  

 

Horizontal translations

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

In this diagram above, the image is a translation of $7$7 units right. We can look at the coordinates of the preimage and image.

Preimage Image
$\left(-5,1\right)$(−5,1) $\left(2,1\right)$(2,1)
$\left(-4,7\right)$(−4,7) $\left(3,7\right)$(3,7)
$\left(0,2\right)$(0,2) $\left(7,2\right)$(7,2)

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.

 

Vertical translations

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

In this diagram, the image is a translation of $5$5 units up. We can look at the coordinates of the preimage and image.

Preimage Image
$\left(1,-3\right)$(1,−3) $\left(1,2\right)$(1,2)
$\left(3,-1\right)$(3,−1) $\left(3,4\right)$(3,4)
$\left(7,-6\right)$(7,−6) $\left(7,-1\right)$(7,−1)

 

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.

 

Combined translations

An object can be translated both horizontally and vertically. 

Exploration

Use the red sliders at the bottom to translate the object. Move the vertices on the original Object to change the shape of the triangle. 

If you are given the coordinates of the Image, what information do you need to find the coordinates of the Preimage Object and vice versa?

Created with Geogebra

 

Practice questions

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 trapezoid $ABCD$ABCD to the trapezoid $A'B'C'D'$A′B′C′D′?

Loading Graph...

A Coordinate Plane is presented with x- and y-axes ranging from -10 to 10. On the plane, there are two shaded trapezoids. One trapezoid 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 trapezoid is labeled $A'$A′, $B'$B′, $C'$C′, and $D'$D′, with similar positioning of the vertices relative to the trapezoid 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

Point $A$A is translated $3$3 units down and $4$4 units to the right, where it now overlaps point $B$B$\left(5,-2\right)$(5,−2).

  1. What were the original coordinates of point $A$A?

    $A$A$=$=$\left(\editable{},\editable{}\right)$(,)

 

Outcomes

I.G.CO.2

Represent transformations in the plane using, e.g. Transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g. Translation versus horizontal stretch).

I.G.CO.4

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

I.G.CO.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g. Graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

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