6. Transformations

Lesson

We have previously discussed congruence transformations. We saw that reflections, rotations, and translations resulted in an image congruent to the preimage, Because congruence holds for these transformations, so does similarity because all congruent figures can be considered similar with a ratio of $1:1$1:1. are also congruent.

Dilations, on the other hand, will result in an image which is similar to the preimage object but is not congruent. Note that not all similar figures are congruent, only those that have a ratio of$1:1$1:1.

We can stretch or compress every point on an object according to the same ratio to perform a dilation. Below is an example of dilating the smaller triangle by a scale factor of $2$2 from the center of enlargement $\left(1,0\right)$(1,0).

Summary

For a dilation using the origin, $\left(0,0\right)$(0,0), as the center with dilation factor $a$`a`, the point $A$`A`$\left(x,y\right)$(`x`,`y`) iis transformed to the point $A'$`A`′$\left(ax,ay\right)$(`a``x`,`a``y`)

Consider the figures shown.

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Are the two triangles congruent, similar or neither?

Congruent

ASimilar

BNeither

CCongruent

ASimilar

BNeither

CWhat is the transformation from triangle $ABC$

`A``B``C`to triangle $A'B'C'$`A`′`B`′`C`′?Dilation

AReflection

BRotation

CTranslation

DDilation

AReflection

BRotation

CTranslation

DWhat is the scale factor of the dilation from triangle $ABC$

`A``B``C`to triangle $A'B'C'$`A`′`B`′`C`′?

Consider the quadrilateral with vertices at $A$`A`$\left(-3,-3\right)$(−3,−3), $B$`B`$\left(-3,3\right)$(−3,3), $C$`C`$\left(3,3\right)$(3,3) and $D$`D`$\left(3,-3\right)$(3,−3), and the quadrilateral with vertices at $A'$`A`′$\left(-9,-9\right)$(−9,−9), $B'$`B`′$\left(-9,9\right)$(−9,9), $C'$`C`′$\left(9,9\right)$(9,9) and $D'$`D`′$\left(9,-9\right)$(9,−9).

Are the two rectangles similar, congruent or neither?

congruent

Asimilar

Bneither

Ccongruent

Asimilar

Bneither

CWhat is the transformation from rectangle $ABCD$

`A``B``C``D`to rectangle $A'B'C'D'$`A`′`B`′`C`′`D`′?dilation

Areflection

Brotation

Ctranslation

Ddilation

Areflection

Brotation

Ctranslation

DWhat is the scale factor of the dilation of rectangle $ABCD$

`A``B``C``D`to rectangle $A'B'C'D'$`A`′`B`′`C`′`D`′?

The quadrilateral with vertices at $\left(9,9\right)$(9,9), $\left(0,9\right)$(0,9), $\left(0,0\right)$(0,0) and $\left(9,0\right)$(9,0) is rotated 90 degrees clockwise around the origin and dilated by a factor of 2 with the origin as the center of dilation.

What are the new coordinates of the vertices of the quadrilateral?

Write all four coordinates on the same line, separated by commas.

What is the sequence of transformations from triangle $ABC$`A``B``C` to triangle $A''B''C''$`A`′′`B`′′`C`′′? Use triangle $A'B'C'$`A`′`B`′`C`′ as a guide.

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translation of 3 units right and 5 units up and rotation of 90 degrees clockwise around the origin

Adilation by a factor of 2 with the origin as the center of dilation and rotation of 90 degrees clockwise around the origin

Bdilation by a factor of 2 with the origin as the center of dilation, and translation of 3 units right and 5 units up

Creflection about the x-axis and dilation by a factor of 2 with the origin as the center of dilation

Dtranslation of 3 units right and 5 units up and rotation of 90 degrees clockwise around the origin

Adilation by a factor of 2 with the origin as the center of dilation and rotation of 90 degrees clockwise around the origin

Bdilation by a factor of 2 with the origin as the center of dilation, and translation of 3 units right and 5 units up

Creflection about the x-axis and dilation by a factor of 2 with the origin as the center of dilation

D

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. Describe a sequence that exhibits the similarity between two given similar figures.