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).
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)
Consider the figures shown.
Are the two triangles congruent, similar or neither?
What is the transformation from triangle $ABC$ABC to triangle $A'B'C'$A′B′C′?
What is the scale factor of the dilation from triangle $ABC$ABC 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?
What is the transformation from rectangle $ABCD$ABCD to rectangle $A'B'C'D'$A′B′C′D′?
What is the scale factor of the dilation of rectangle $ABCD$ABCD 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 $ABC$ABC to triangle $A''B''C''$A′′B′′C′′? Use triangle $A'B'C'$A′B′C′ as a guide.
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.