New York Geometry - 2020 Edition
4.01 Translations
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?

#### Practice questions

##### Question 1

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

1. $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

##### Question 2

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

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

$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 point $A$A, at some initial position. 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 was the coordinate of the original point $A$A?

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

### Outcomes

#### GEO-G.CO.2

Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not.

#### GEO-G.CO.4

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

#### GEO-G.CO.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another.