1. Investigating Properties of Parallelism and the Center
Adapted from Walch Education

2. Dilations Dilations require a center of dilation and a scale factor.
The center of dilation is the point about which all points are stretched or compressed.
The scale factor of a figure is a multiple of the lengths of the sides from one figure to the transformed figure.
Side lengths are changed according to the scale factor, k.
The scale factor can be found by finding the distances of the sides of the preimage in relation to the image.
1.1.1: Investigating Properties of Parallelism and the Center

3. Scale Factor
Use a ratio of corresponding sides to find the scale factor:
The scale factor, k, takes a point P and moves it along a line in relation to the center so that
1.1.1: Investigating Properties of Parallelism and the Center

4. 1.1.1: Investigating Properties of Parallelism and the Center

5. Key Concepts
If the scale factor is greater than 1, the figure is stretched or made larger and is called an enlargement. (A transformation in which a figure becomes larger is also called a stretch.)
If the scale factor is between 0 and 1, the figure is compressed or made smaller and is called a reduction. (A transformation in which a figure becomes smaller is also called a compression.)
If the scale factor is equal to 1, the preimage and image are congruent. This is called a congruency transformation.
1.1.1: Investigating Properties of Parallelism and the Center

6. Key Concepts, continued
Angle measures are preserved in dilations.
The orientation is also preserved.
The sides of the preimage are parallel to the corresponding sides of the image.
The corresponding sides are the sides of two figures that lie in the same position relative to the figures.
1.1.1: Investigating Properties of Parallelism and the Center

7. Key Concepts, continued
In transformations, the corresponding sides are the preimage and image sides, so and
are corresponding sides and so on.
The notation of a dilation in the coordinate plane is given by Dk(x, y) = (kx, ky). The scale factor is multiplied by each coordinate in the ordered pair.
The center of dilation is usually the origin, (0, 0).
1.1.1: Investigating Properties of Parallelism and the Center

8. Key Concepts, continued
If a segment of the figure being dilated passes through the center of dilation, then the image segment will lie on the same line as the preimage segment. All other segments of the image will be parallel to the corresponding preimage segments.
The corresponding points in the preimage and image are collinear points, meaning they lie on the same line, with the center of dilation.
1.1.1: Investigating Properties of Parallelism and the Center

9. 1.1.1: Investigating Properties of Parallelism and the Center

10. Properties of Dilations
Shape, orientation, and angles are preserved.
All sides change by a single scale factor, k.
The corresponding preimage and image sides are parallel.
The corresponding points of the figure are collinear with the center of dilation.
1.1.1: Investigating Properties of Parallelism and the Center

11. Thanks for Watching!
Ms. Dambreville