Unit 3, Lesson 3
Measuring Dilations
Geometry
3.1
Warm-up
Instructional Routines
NoneMaterials
To Gather
Rulers marked with centimetersActivity Narrative
In this Warm-up, students dilate a figure with the center of dilation in the interior of the figure. This slight difference should allow students to practice using precision with their understanding of dilations.
Launch
NoneActivity
NoneDilate triangle using center and a scale factor of 3.
Student Response
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Building on Student Thinking
Activity Synthesis
Display the image, and ask students why it is incorrect. (The image of should be on the same side of as . You can tell it's wrong because the triangles don't look like scaled copies.)
Make sure that all students understand how to draw rays from the center point through the points to be dilated, and find the image of the points by measuring along the rays. Students will have more opportunities to dilate figures in other activities in this lesson, so there is no need to have them correct any errors they made.
Lesson Synthesis
Display this image:
Ask students where they would place points along ray in order to make a scaled copy of the triangle with different scale factors. Record their thinking for all to see.
- “If you wanted to make a scaled copy with side lengths twice the size of the original, where should you place ? Explain how you know.” (16 units from along ray , or 8 units past along ray . In order for the scaled copy to be twice as big, it has to be twice as far from the center. So because is 8 units from , make 16 units from .)
- “If you wanted to make a scaled copy with side lengths half the size of the original, where should you place ? Explain how you know.” (4 units from along ray . In order for the scaled copy to be half as big, it has to be half as far from the center. So because is 8 units from , make 4 units from .)
- “If you wanted to make a scaled copy with side lengths the size of the original, where should you place ?” (2 units from along ray .)
- “If you wanted to make a scaled copy with side lengths the size of the original, where should you place ?” (6 units from along ray .)
Invite students to explain what they learned about dilations and specifically what can be determined from knowing the distance to . (The ratio of the distances from the original to the center and the scaled copy to the center is the same as the scale factor from the original to the scaled copy.)
Student Lesson Summary
We know that a dilation with center and positive scale factor, , takes a point along the ray to another point whose distance is times farther away from than is.
The triangle is a dilation of the triangle with center and with a scale factor of 2. So is 2 times farther away from than is, is 2 times farther away from than is, and is 2 times farther away from than is.
Because of the way dilations are defined, all of these quotients give the scale factor: .
If triangle is dilated from point with scale factor , then each vertex in is on the ray from P through the corresponding vertex of , and the distance from to each vertex in is one-third as far as the distance from to the corresponding vertex in .
The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor. In other words, if segment is dilated from point with a scale factor of , then the length of segment is multiplied by to get the corresponding length of .
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Corresponding side lengths of the original figure and dilated image are all in the same proportion, and are related by the same scale factor, .