Lesson 4 | Dilations and Similarity | 10th Grade Mathematics

Objective

Divide a line segment into equal sections using dilation.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.CO.D.12 — Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Congruence

G.CO.D.12 — Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G.SRT.A.1.A — 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.

Similarity, Right Triangles, and Trigonometry

G.SRT.A.1.A — 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.
G.SRT.A.1.B — The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Similarity, Right Triangles, and Trigonometry

G.SRT.A.1.B — The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Foundational Standards

The foundational standards covered in this lesson

8.G.A.1

Geometry

8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe that a dilation of a line not passing through the center of the dilation will map to a parallel line.
Use the property of dilations with respect to parallel lines to describe that the benchmark line segment must be parallel to the line segment that is to be partitioned.
Construct a line parallel to the original line.
Use vectors from the center of dilation and the benchmark points of segmentation to dilate onto the segment to be partitioned.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Dilate the line segment $${\overline{ST}}$$ by $$2$$ from center point $$R$$. Label the new line segment as $${\overline{KM}}$$.

Guiding Questions

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Problem 2

Divide segment $${AB}$$ into three segments of equal length.

Below is line segment $${{AB}}$$ and line $${DG}$$, which is parallel to $${{AB}}$$. Points $$D$$, $$E$$, $$F$$, and $$G$$ are all equidistant from one another using the same compass setting.

If points $$D$$ and $$G$$ are dilations of points $$A$$ and $$B$$ respectively, how can you use the points $$E$$, $$F$$, and $$G$$ to partition line segment $${{AB}}$$?

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Divide the following line segment into a ratio of 1:4 using dilations. Then, describe the parallel relationship you have created through this process.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students are practicing constructing parallel lines from Unit 1.
Include problems where students need to divide a line segment into a ratio of ____. Ensure that students identify the total number of equal pieces that need to be identified to do this correctly.
Include problems where the construction is done for the students and steps are provided. Ask students to identify and justify the relationships in the diagram.
Include problems where the construction is given and students need to provide the steps.
Include problems where a fictitious student has made an error or has gotten stuck in partitioning a segment. Ask students to identify and correct the error. Focus on errors where the number of equal segments is not adequate for the ratio given.

EngageNY Mathematics Geometry > Module 2 > Topic B > Lesson 10 — Focus on the dilation method, not the side splitter method.

Lesson 3

Lesson 5

Lesson Map

Topic A: Dilations off the Coordinate Plane

Describe properties of scale drawings.

G.SRT.A.2

Define and describe the characteristics of dilations. Dilate figures using constructions when the center of dilation is not on the figure.

G.CO.A.2 G.SRT.A.2 G.SRT.A.3

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

G.SRT.A.2

Divide a line segment into equal sections using dilation.

G.CO.D.12 G.SRT.A.1.A G.SRT.A.1.B

Dilate a figure from a point on the figure. Use the properties of dilations with respect to parallel lines to verify dilations and find the center of dilation.

G.CO.D.12 G.SRT.A.1.A G.SRT.A.2

Prove that a line parallel to one side of a triangle divides the other two sides proportionally.

G.CO.C.10 G.SRT.B.4 G.SRT.B.5

Identify measurements in dilated figures with the center of dilation on the figure directly and algebraically.

G.SRT.A.2

Topic B: Dilations on the Coordinate Plane

Dilate a figure on the coordinate plane when the center of dilation is the origin.

G.CO.A.2 G.SRT.A.2

Dilate a figure when the center of dilation is not the origin. Determine center of dilation given the original and dilated figure.

G.CO.A.2 G.SRT.A.2

Topic C: Defining Similarity

Define similarity transformation as the composition of basic rigid motions and dilations. Describe similarity transformation applied to an arbitrary figure.

G.SRT.A.2 G.SRT.B.5

Prove that all circles are similar.

G.C.A.1

Prove angle-angle criterion for two triangles to be similar.

G.SRT.A.3

Use angle-angle criterion to prove two triangles to be similar.

G.SRT.A.3

Develop the side splitter theorem and side-angle-side similarity criteria, and use these in the solution of problems.

G.SRT.B.5

Topic D: Similarity Applications

Develop the angle bisector theorem based on facts about similarity and congruence, and use this in the solution of problems.

G.SRT.B.5

Use the side-side-side criteria for similarity and other similarity and congruence theorems in the solution of problems.

G.SRT.B.5

Solve for measurements involving right triangles using scale factors and ratios.

G.SRT.B.5

Solve real-life problems with two different centers of dilation.

G.SRT.B.4 G.SRT.B.5

Dilations and Similarity

Objective

Common Core Standards

Core Standards

Congruence

Similarity, Right Triangles, and Trigonometry

Similarity, Right Triangles, and Trigonometry

Foundational Standards

Geometry

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Additional Practice

Lesson Map

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