Objective: When done with this lesson, you will have demonstrated how to
predict changes in perimeter, lengths, angles, and areas under a dilation
find the center of dilation when an original figure and its image under a dilation are given
find the scale factor of a dilation when an original figure and its image under a dilation are given
determine if polygons are "similar"
Washington State GLE:
Approximate completion time: 3 hours
In "Dilations - part 1" you investigated two different translations.
The rule P(x, y) → P'(x, 3y) stretches a shape by a factor of 3 only in the vertical direction. This is a distortion.
The rule P(x, y) → P'(3x, 3y) stretches a shape by a factor of 3 in both the vertical and horizontal directions. This is a dilation.
A dilation is a transformation that changes the size of an object without changing the shape.
In a dilation, the following always happen:
there is a center of dilation
there is a scale factor
angles do not change measure
corresponding lengths are proportionate
Any shape and its image under a dilation are called similar.
Consider this dilation and what happens.
Do you see the center of dilation? It doesn't have to be at the origin, it can be anywhere.
Do you see the lengths of the original pentagon grow by a factor of 4 into the lengths in the image? That means the scale factor is 4.
(Note: the square root values are exact lengths which we can find many ways, including the pythagorean theorem.)
Do you see the corresponding angles do not change in their measure?
Do you the proportions of corresponding sides?
(The left side of the original compares to the left side of the image as does the bottom side of the original to the bottom side of the image.
Which is written as 2:8 = 3:12 or 2/8 = 3/12.)
Do you notice anything else in the dilation above? What about the perimters, areas, diagonals, and orientation?
You will investigate what happens to many of these things in your assignment.
Fun dilation extras:
Drawing dilations on a shape can create a 3D feel or look.
Dilations can be combined with other transformations to help create many effects.
Click on the bee below to watch an multiple transformations working simultaneously.
