Dilations and Similarity

Students use constructions to explore dilations in order to define and establish similarity, and they prove and use similarity criterion and theorems in the solution of problems.

Unit Summary

In Unit 3, Dilations & Similarity, students contrast the properties of rigid motions to establish congruence with dilations, a non-rigid transformation to establish similarity. Constructions are again used to reveal the properties of dilations and partition figures into proportional sections. Students discover additional relationships within and between triangles using proportional reasoning. The topics in this unit serve as the underpinning for trigonometry studied in Unit 4 and provide the first insight into geometry as a modeling tool for contextual situations. 

This unit begins with Topic A, Dilations off the Coordinate Plane. Students identify properties of dilations by performing dilations using constructions. Students use appropriate tools and also look for regularity in their constructions to draw conclusions. Students are familiar with some of the conceptual ideas around dilations from their work in eighth grade to compare and contrast dilations with rigid motions. In this topic, students develop the dilation theorem- important for establishing additional reasoning for triangle congruence in the next topic. 

Topic B formalizes coordinate point relationships with dilations on the coordinate plane. Students relate their understanding of dilations off the coordinate plane to dilations on the coordinate plane both using the origin as a center of dilation and using other points on the coordinate plane as the center of dilation. 

In Topic C, students formalize the definition of “similarity,” explaining that the use of dilations and rigid motions are often both necessary to prove similarity. Students develop triangle similarity criteria and the side splitter theorem, using them to solve for missing measures and angles in mathematical and real-world problems. Students also discover that all circles are similar in this topic. 

Students will use similarity theorems and relationships to establish additional relationships with trigonometric ratios in the next unit.

Assessment

This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

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Internalization of Standards via the Unit Assessment

  • Take unit assessment. Annotate for: 
    • Standards that each question aligns to
    • Purpose of each question: spiral, foundational, mastery, developing
    • Strategies and representations used in daily lessons
    • Relationship to Essential Understandings of unit 
    • Lesson(s) that assessment points to

Internalization of Trajectory of Unit

  • Read and annotate "Unit Summary."
  • Notice the progression of concepts through the unit using “Unit at a Glance.”
  • Do all target tasks. Annotate the target tasks for: 
    • Essential understandings
    • Connection to assessment questions

Essential Understandings

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  • Rigid motions and dilations are used to prove that two figures are similar, and they are the basis for developing triangle similarity criteria. 
  • The properties of dilations describe parallel relationships between corresponding line segments, collinear relationships between points, proportional relationships between lengths of corresponding line segments, and congruent relationships between corresponding angle measures. 
  • The side splitter theorem, dilation theorem, and triangle similarity criteria can be used to prove and identify relationships in geometric figures. 

Vocabulary

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similar scale factor dilation
center of dilation Side-side-side (SSS) similarity proportionality
vector Dilation theorem Side splitter theorem
Angle-angle (AA) similarity Side-angle-side (SAS) similarity Angle bisector theorem
corresponding parts    

 

Lesson Map

Topic A: Dilations off the Coordinate Plane

Topic B: Dilations on the Coordinate Plane

Topic C: Defining Similarity

Topic D: Similarity Applications

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Circles
  • G.C.A.1 — Prove that all circles are similar.

Congruence
  • G.CO.A.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).

  • G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

  • G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

  • 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.

Similarity, Right Triangles, and Trigonometry
  • G.SRT.A.1 — Verify experimentally the properties of dilations given by a center and a scale factor:

  • 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.

  • 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.

  • G.SRT.A.3 — Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

  • G.SRT.B.4 — Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

  • G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Foundational Standards

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Geometry
  • 7.G.A.1

  • 7.G.A.2

  • 8.G.A.1

  • 8.G.A.2

  • 8.G.A.3

  • 8.G.A.4

  • 8.G.A.5

Future Standards

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Circles
  • G.C.B.5

Similarity, Right Triangles, and Trigonometry
  • G.SRT.C.6

  • G.SRT.C.7

  • G.SRT.C.8

Trigonometric Functions
  • F.TF.A.1

  • F.TF.A.2

  • F.TF.A.3

  • F.TF.A.4

  • F.TF.B.5

  • F.TF.B.6

  • F.TF.B.7

  • F.TF.C.8

  • F.TF.C.9

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.