Dilations and Similarity

Lesson 3

Math

Unit 3

10th Grade

Lesson 3 of 18

Objective


Verify that dilations result in congruent angles and proportional line segments.

Common Core Standards


Core Standards

  • G.SRT.A.2 — 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.

Foundational Standards

  • 8.G.A.1

Criteria for Success


  1. Identify that a dilation maps a ray to a ray and a line segment to a line segment, sending the endpoint to the endpoint. 
  2. Describe that regardless of scale factor, or center of dilation, angles in the original figure will always be congruent to angles in the dilated figure. 
  3. Explain that the relationship between the ratio of each side length in the original figure will be proportional to the corresponding side length in the dilated figure. 
  4. Explain that a geometric figure will maintain all attributes except size when dilated (i.e., a ray will map to a ray, a line segment will map to a line segment, a point will map to a point, a square will map to a square, a circle will map to a circle). 
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Anchor Problems

25-30 minutes


Problem 1

Identify if the following statement is always, sometimes, or never true. Provide examples or criteria that support your reasoning. 

“A dilation of an angle marked by two line segments will stretch or shrink the measure of the angle by the scale factor."

Guiding Questions

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Problem 2

Describe whether each statement is true or false. 

  1. When a line segment is dilated, the dilated figure is a congruent line segment. 
  2. When a point is dilated, the dilated figure is a larger point. 
  3. When a ray is dilated, the result is a ray.

Guiding Questions

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Problem 3

Below is rectangle $$A$$ that has been dilated by a scale factor of 1.5 to form rectangle $$B$$
Find the length of the diagonal in rectangle $$B$$

 

Guiding Questions

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Target Task

5-10 minutes


Below is a figure, $$C$$, and its dilation, $$D$$, by a factor of $${\frac{4}{3}}$$. 

  • Describe what you know about the corresponding angles based on the fact that the figures are dilations of one another. 
  • Describe what you know about the corresponding sides (using correct mathematical notation) based on the fact that the figures are dilations of one another. 
  • Give the scale factor to dilate figure $$D$$ that will result in figure $$C$$. 

Next

Divide a line segment into equal sections using dilation.

Lesson 4
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Lesson Map

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Topic A: Dilations off the Coordinate Plane

Topic B: Dilations on the Coordinate Plane

Topic C: Defining Similarity

Topic D: Similarity Applications

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