Triangle is a dilation of triangle using center and a scale factor of 2.
What is the same about the two triangles? What is different? Make a conjecture about what stays the same after dilation.
Use the tools available to figure out if what you thought was true is definitely true for these triangles.
Do you think your conjecture will be true for any figure after dilation?
4.2
Activity
Dilating Lines
Standards Alignment
Building On
G-CO.2
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Jada dilates triangle using center and a scale factor of 2.
Prove the conjecture from your whole-class discussion.
In Jada’s diagram the scale factor is greater than one. Would your proof have to change if the scale factor is less than one?
Student Lesson Summary
When one figure is a dilation of the other, we know that corresponding side lengths of the original figure and the dilated image are in the same proportion, and are all related by the same scale factor, . What is the relationship of corresponding angles in the original figure and the dilated image?
For example, if triangle is dilated, using center , with a scale factor of 2, we can verify experimentally that each angle in triangle is congruent to its corresponding angle in triangle . is congruent to . is congruent to . is congruent to .
What is the image of a line not passing through the center of dilation? For example, what will be the image of line when it is dilated with center and a scale factor of 2? We can use congruent corresponding angles to show that line is taken to parallel line .
What is the image of a line passing through the center of dilation?
For example, what will be the image of line when it is dilated with center and a scale factor of ? When line is dilated with center and a scale factor of , line is unchanged, because dilations take points on a line through the center of a dilation to points on the same line, by definition.
So 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.
Glossary
None
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Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.