Lesson

Using three forms of transformations, Rotations, Reflections and Translations, we can create congruent shapes. In fact all pairs of congruent shapes can be matched to each other using a series or one or more of these three transformations.

We can reflect every point on the preimage object in a line to get our transformed image. Below is an example reflecting over the $y$`y`-axis. What do you notice about the coordinates of the given point when they undergo a reflection? When it is reflected over the $y$`y`-axis the $y$`y`-value stays the same, but the $x$`x`-value is negated.

Summary

If the point $A$`A`$\left(a,b\right)$(`a`,`b`) is reflected over the $y$`y`-axis it is transformed to the point $A'$`A`′$\left(-a,b\right)$(−`a`,`b`)

If the point $A$`A`$\left(a,b\right)$(`a`,`b`) is reflected over the $x$`x`-axis it is transformed to the point $A'$`A`′$\left(a,-b\right)$(`a`,−`b`)

A shape is rotated around a center point in a circular motion. Focus on the vertex opposite the shortest side, $\left(3,1\right)$(3,1). What happens to the coordinates when the shape is rotated $90^\circ$90° clockwise? It has rotated one quadrant, so the sign changes for one of the values (in this case the $y$`y`-value) and the rotation has swapped the coordinates to give us $\left(1,-3\right)$(1,−3).

Summary

If the point $A$`A`$\left(a,b\right)$(`a`,`b`) is rotated $90^\circ$90° clockwise about the origin it is transformed to the point $A'$`A`′$\left(-b,a\right)$(−`b`,`a`)

If the point $A$`A`$\left(a,b\right)$(`a`,`b`) is rotated $180^\circ$180° clockwise about the origin it is transformed to the point $A'$`A`′$\left(-a,-b\right)$(−`a`,−`b`)

If the point $A$`A`$\left(a,b\right)$(`a`,`b`) is rotated $270^\circ$270° clockwise about the origin it is transformed to the point $A'$`A`′$\left(b,-a\right)$(`b`,−`a`)

The whole shape moves the same distance in the same direction, without being rotated or flipped. In the picture below, we can see the object has been moved up $5$5 units.

Summary

If the point $A$`A`$\left(a,b\right)$(`a`,`b`) is translated $k$`k` up, it is transformed to the point $A'$`A`′$\left(a,b+k\right)$(`a`,`b`+`k`)

If the point $A$`A`$\left(a,b\right)$(`a`,`b`) is translated $h$`h` units right, it is transformed to the point $A'$`A`′$\left(a+h,b\right)$(`a`+`h`,`b`)

While reflections, rotations and translations resulted in an image congruent to the preimage, dilations will result in an image which is similar to the preimage object.

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.

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.

Given a regular or irregular polygon, describe the rotations and reflections (symmetries) that map the polygon onto itself.

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Verify experimentally the properties of dilations given by a center and a scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. Explain using similarity transformations that similar triangles have equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.