In middle school, students confirmed experimentally some of the properties of dilations (dilations take angles to congruent angles and lines to parallel lines). In this lesson, students verify experimentally and assert that dilations take angles to congruent angles, and use this assertion to create a convincing argument that if dilations take angles to congruent angles, they must take lines to parallel lines (MP3). Students draw on their work with angles formed by parallel lines and transversals in an earlier unit to complete the proof.
Students also examine what happens to lines through the center of dilation under dilation. They reason based on the definition of dilation that those lines don’t change under dilation. Verifying the properties of dilation prepares students to use those properties in proofs involving similarity, such as proving the Angle-Angle Triangle Similarity Theorem.
- Comprehend that dilations take angles to congruent angles.
- Prove that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged (orally and in writing).
- Let’s dilate lines and angles.
Prepare additional copies of the Blank Reference Chart blackline master (double sided, 1 per student). Students can staple the new chart to their full ones as they will need to continue to refer to the whole packet.
- I can explain what happens to lines and angles in a dilation.
A dilation with center \(P\) and positive scale factor \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.
Triangle \(A'B'C'\) is the result of applying a dilation with center \(P\) and scale factor 3 to triangle \(ABC\).
The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.