written assignment 7 dilations and symmetry answer key

Math  /  10th Grade  /  Unit 3: Dilations and Similarity

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, 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 8th 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 .

Pacing: 20 instructional days (18 lessons, 1 flex day, 1 assessment day)

The following assessments accompany Unit 3.

Use the resources below to assess student understanding of the unit content and action plan for future units.

Post-Unit Assessment

Post-Unit Assessment Answer Key

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Unit Assessment

• 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.”
• Essential understandings
• Connection to assessment questions

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

• 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. 

Terms and notation that students learn or use in the unit

Topic A: Dilations off the Coordinate Plane

Describe properties of scale drawings.

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.

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.

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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.

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.

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

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

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

Topic D: Similarity Applications

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

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

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

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

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

Common Core Standards

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

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

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.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

Standards covered in previous units or grades that are important background for the current unit

7.G.A.1 — Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

7.G.A.2 — Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

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

8.G.A.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

8.G.A.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

8.G.A.4 — Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

8.G.A.5 — Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Future Standards

Standards in future grades or units that connect to the content in this unit

G.C.B.5 — Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

G.SRT.C.6 — Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G.SRT.C.7 — Explain and use the relationship between the sine and cosine of complementary angles.

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Trigonometric Functions

F.TF.A.1 — Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

F.TF.A.2 — Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

F.TF.A.3 — Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.

F.TF.A.4 — Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

F.TF.B.5 — Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

F.TF.B.6 — Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

F.TF.B.7 — Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

F.TF.C.8 — Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

F.TF.C.9 — Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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.

Congruence in Two Dimensions

Right Triangles and Trigonometry

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Dilations and Symmetry Review Notes For Test

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Unit 7 – Dilations and Similarity

LESSON/HOMEWORK

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

SMART NOTEBOOK

Dilations in the Coordinate Plane

Dilations and Angles

Similar Figures

Similarity Criteria

Reasoning with Similarity

More Similarity Reasoning

The Side Splitter Theorem

Partitioning a Line Segment

The Medians of a Triangle

Partitioning the Hypotenuse with an Altitude

Geometric Means

Proving the Pythagorean Theorem (No Homework)

Unit Review

Unit 7 Review

UNIT REVIEW

EDITABLE REVIEW

Unit 7 Assessment – Form A

EDITABLE ASSESSMENT

Unit 7 Exit Tickets

U07.AO.01 – Partitioning a Side with an Angle Bisector (Extended Activity)

EDITABLE RESOURCE

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