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Transformations Rules

Translations in the Coordinate Plane
Translation (x, y) →(x ± a, y ± b)

a represents the number of units the figure will be translated left or right.

b represents the number of units the figure will be translated up or down.

Reflections Rules

Reflections in the Coordinate Plane
Reflection across the x-axis (x, y) → (x, −y)
Reflection across the y-axis (x, y) → (−x, y)
Reflection across the line y = x (x, y) → (y, x)

Rotation Rules

Rotations in the Coordinate Plane (clockwise)
Rotation 90° (x, y) → (y, −x)
Rotation 180° (x, y) → (−x, −y)
Rotation 270° (x, y) → (−y, x)

Compositions of Transformations

A composition of transformations is a transformation that is formed from a combination of transformations. In other words, it a transformation followed by other transformations.

Three images. Left: Triangle ABC with caption 'preimage'. Center: Triangle A'B'C', with an angle showing a rotation swinging the triangle around about 120 degrees. The right image is the rotated triangle A''B''C'', turned 120 degrees clockwise from its original position.

The diagram shows a composition of transformations in which ∆ABC is translated to the right and then rotated.

Double Prime Notation

Anytime you have three figures of a composition of transformations, the second figure is labeled as primes after the letters. Prime notation has an apostrophe (') after the letters. The third figure is labeled as double primes after the letters. Double prime notation has two apostrophes ('') after the letters.

For example, in the image below, if the first figure is ABC, then the second figure is A’B’C’ and the third figure is A”B”C”. ABC is the preimage and A”B”C” is the image.

Three images. Left: Triangle ABC with caption 'preimage'. Center: Triangle A'B'C', with an angle showing a rotation swinging the triangle around about 120 degrees. The right image is the rotated triangle A''B''C'', turned 120 degrees clockwise from its original position.

Compositions of Two Translations

A composition of two translations is a translation after a translation. The final image represents the composition of the translations. The composition of two translations is the same as a translation of the preimage.

To get the vertical transformation of the preimage to the image, you will need to find the sum of the two vertical translations.

To get the horizontal translation of the preimage to the image, you will need to find the sum of the two horizontal translations.

Triangle ABC, with arrows marking its first translation counterclockwise, followed by a second translation counterclockwise again; A begins at the top left corner, A' is at the bottom, then A'' is at the top right. A red arrow shows that given the two translations counterclockwise are the same as one translation to clockwise.

In the diagram, point A is translated 3 units right and 6 units down. Then, translated 4 units right and 5 units up. The image A”B”C” is the result of the two translations, the composition of the transformations.

Triangle ABC, with arrows marking its first translation counterclockwise, followed by a second translation counterclockwise again; A begins at the top left corner, A' is at the bottom, then A'' is at the top right. A red arrow shows that given the two translations counterclockwise are the same as one translation to clockwise.

Note: The composition of the two translations is a translation of the preimage 7 units right and 1 unit down. Find the sum of the two horizontal translations ((+3) + (+4)) to get the horizontal translation (+7) of the preimage to the image. Find the sum of the two vertical translations ((−6) + (+5)) to get the vertical translation (−1) of the preimage to the image.

Glide Reflections

A glide reflection is an image that is formed by a translation of a figure, followed by a reflection of the translation of the figure. It can also be an image that is formed by a reflection of a figure, followed by a translation of the reflection of the figure.

Triangle ABC being translated to the right, so that it is in the same configuration but shifted right; and then mirrored around the y axis, causing it to be exactly mirrored to the right.

Example #1

Watch Graph a Composition of Transformations.

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Example #2

Watch Graph a Composition of Transformations.

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Reflections Over Two Parallel Lines Theorem

The Reflection Over Two Parallel Lines Theorem states that:

A reflection of a figure over a line followed by a reflection over a line that is parallel to that line is the same as a one translation of the figure.

Triangle DEF reflected across the y axis to become reversed, and then reflected across the y axis again to become reversed again. The final triangle D''E''F'' is in the same orientation as the original triangle DEF.

A reflection of a figure over a line followed by a reflection over a line that is parallel to that line, followed by another reflection over a third line is the same as a one reflection of the figure.

Triangle DEF being reflected three times, ending with triangle D'''E'''F''' being the reverse of the original triangle's configuration.

Reflections Over Intersecting Lines Theorems

The Reflection Over Two Intersecting Lines Theorem states that a composition of reflections of a figure over two intersecting lines is the same as a one rotation of the figure.

Triangle XYZ reflected over the y axis, with triangle X'Y'Z' being a mirror reflection; triangle X'Y'Z' is then reflected over a diagonal line, flipping triangle X''Y''Z'' into a different configuration.

Application

Composition of Transformations and Congruence

The rigid motions that hold congruence are translations, reflections, and rotations. If an image is the result of a rigid transformation or a composition of rigid transformations on a figure, then it is congruent to the original figure (preimage).

Triangle DEF is reflected over the y axis twice. The units making up each of its sides are first reversed, then set back to their initial configuration.

Example #3

Watch Determine Congruence Using Transformations.

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