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Informal Proofs of Properties of Dilations




 


Videos and solutions to help Grade 8 students learn an informal proof of why dilations are degree-preserving transformations.



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Common Core For Grade 8



New York State Common Core Math Module 3, Grade 8, Lesson 7


Lesson 7 Student Outcomes


• Students know an informal proof of why dilations are degree-preserving transformations.
• Students know an informal proof of why dilations map segments to segments, lines to lines, and rays to rays.

Lesson 7 Summary


• We know an informal proof for dilations being degree-preserving transformations that uses the definition of dilation, the Fundamental Theorem of Similarity, and the fact that there can only be one line through a point that is parallel to a given line.
• We informally verified that dilations of segments map to segments, dilations of lines map to lines, and dilations of rays map to rays.


NYS Math Module 3 Grade 8 Lesson 7


Classwork
Discussion
Exercise
Use the diagram below to prove the theorem: Dilations preserve the degrees of angles.
Let there be a dilation from center O with scale factor r. Given ∠PQR, show that P' = dilation(P), Q' = dilation(Q) and R' = dilation(R), then |∠PQR| = |∠P'Q'R'|. That is, show that the image of the angle after a dilation has the same measure, in degrees, as the original.

Example 1
In this example, students verify that dilations map lines to lines.

Example 2
In this example, students verify that dilations map segments to segments.

Example 3
In this example, students verify that dilations map rays to rays.






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