Pennsylvania 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 $xy$`x``y`-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 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.

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.

An object can be translated both horizontally and vertically.

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?

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 trapezoid $ABCD$`A``B``C``D` to the trapezoid $A'B'C'D'$`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 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).

What was the coordinate of the original point $A$

`A`?$A$

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

Use geometric figures and their properties to represent transformations in the plane.