8.02 Similarity transformations
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
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
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
Asimilar
Bneither
CWhat is the transformation from rectangle $ABCD$ABCD to rectangle $A'B'C'D'$A′B′C′D′?
dilation
Areflection
Brotation
Ctranslation
DWhat is the scale factor of the dilation of rectangle $ABCD$ABCD to rectangle $A'B'C'D'$A′B′C′D′?