4.04 Similarity transformations

Lesson

Practice

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.

Dilations (Enlargements)

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).

Image is similar to the preimage

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′$\left(ax,ay\right)$(ax,ay)

Practice questions

Question 1

Consider the figures shown.

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Two triangles are depicted on a Cartesian coordinate plane with x and y-axes ranging from -10 to 10. The larger triangle, labeled with vertices A' $\left(-6,-2\right)$(−6,−2), B' $\left(2,6\right)$(2,6), and C' $\left(6,-6\right)$(6,−6), is shaded in light gray. The smaller triangle, labeled with vertices A $\left(-3,-1\right)$(−3,−1), B $\left(1,3\right)$(1,3), and C $\left(3,-3\right)$(3,−3), is shaded in dark gray and positioned inside the larger triangle. Although they share the same shape, the triangles differ in size

Are the two triangles congruent, similar or neither?
Congruent
A
Similar
B
Neither
C
What is the transformation from triangle $ABC$ABC to triangle $A'B'C'$A′B′C′?
Dilation
A
Reflection
B
Rotation
C
Translation
D
What is the scale factor of the dilation from triangle $ABC$ABC to triangle $A'B'C'$A′B′C′?

Question 2

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
A
similar
B
neither
C
What is the transformation from rectangle $ABCD$ABCD to rectangle $A'B'C'D'$A′B′C′D′?
dilation
A
reflection
B
rotation
C
translation
D
What is the scale factor of the dilation of rectangle $ABCD$ABCD to rectangle $A'B'C'D'$A′B′C′D′?

Question 3

Question 4

Outcomes

NY-8.G.4

Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane.