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

CC.2.3.8.A.2

Understand and apply congruence, similarity, and geometric transformations using various tools.

M08.C-G.1.1.4

Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them.