Similarity and Similarity Transformations

The Lesson Activities will help you meet these educational goals:

  • Content Knowledge—You will use the definition of similarity in terms of similarity transformations to decide whether two given figures are similar.
  • Inquiry—You will perform an investigationin which you will make observations, analyze results, communicate your results in tables and written form, and draw conclusions.
  • 21stCentury Skills—You will employ online tools for research and use critical-thinking and problem-solving skills.

Directions

You will evaluatesome of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-CheckedActivities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sampleresponses to evaluate your own work.

  1. Similarity Transformations

In this activity, you will use GeoGebra to explore using similarity transformations to map one polygon onto another. Go to similaritytransformations,and complete each step below.If you need help, follow these instructions for using GeoGebra.

  1. Determine the sequences of rigid motions and dilations that map the preimage ABCDE to each of its images. If no sequence of rigid motions and dilations maps ABCDE onto an image, state that in the table. Remember to describe each transformation completely. For example, you must specify a line of reflection for each reflection and a center of dilation for each dilation.Perform the transformations in GeoGebra to check your responses.

Type your response here:

Image / Sequence ofRigid Motions and Dilations
FGHIJ
KLMNP
QRSTU
VWXYZ
  1. Based on your work with similarity transformations in part a, which images are similar to the preimage ABCDE? Which images are not similar?

Type your response here:

  1. Often, there is more than one set of sequences that will take a preimage to an image. Determine one ortwo other sequences to create FGHIJ from ABCDE.

Type your response here:

Image / Sequence ofRigid Motions and Dilations
FGHIJ
FGHIJ
  1. Calculate the lengths of thecorresponding sides for FGHIJ and ABCDE. You will need to use the distance formula, for sides that are not purely horizontal or vertical.Show your work in the table. Afterward, confirm your calculations using GeoGebra.

Type your response here:

Side / Calculation / Using GeoGebra
  1. What is the ratio of the side lengths onpolygon FGHIJto the corresponding side lengths on polygonABCDE? Enter the ratio in the table. How does this ratio relate to the scale factor of the dilation?

Type your response here:

Side Lengths / Ratio
FG:AB
GH:BC
HI:CD
IJ:DE
JF:EA
  1. Based on your work in parts a through e, what is a similarity transformation? What is necessary to determine whether two or more shapes are similar? Explain in terms of rigid transformations, dilations, proportionality, and scale factors.

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Similarity for Triangles

In this activity, you will use GeoGebra to explore similarity for triangles. You’ll check whether a similarity transformation exists that will demonstrate that ∆ABC and ∆DEF are similar and record your observations about the two triangles. Go to similarity for triangles,and complete each step below.

  1. Use the distance formula,tocalculatethe ratio of a sidelength on ∆DEFto the corresponding side lengthon ∆ABC. Show your work. Then confirm your calculations using GeoGebra tools.

Type your response here:

  1. Find one or more sequences of rigid motions and dilations that will map ∆ABC to ∆DEF.Often, there is more than one sequence that will map a preimage to a similar image.The table below provides spaces for two different sequences.

Begin by setting the scale factor of each dilation, n, equal to the ratio you computed in part a. Then find the rigid transformations needed to complete the mapping. Perform the transformations in GeoGebra to check your sequences.

Type your response here:

Sequence 1 / Sequence 2
  1. You just completed a similarity transformation to show that ∆ABC is similar to ∆DEF. Let’s see how the corresponding angles relate between these two similar triangles. Measure the angles of∆ABC and ∆DEF and record their values.

Type your response here:

Angle / Measure / Angle / Measure
  1. Compare the corresponding angles in ∆ABC and ∆DEF.In general, what do your observations suggestabout the angle measures in two similar triangles? Use the definition of similarity transformations to explain the relationship between corresponding angle measures.

Type your response here:

  1. Use the distance formula to calculate the length of each line segment of ∆ABC and ∆DEF. (You did this for one line segment in part a.) Determine the ratio of the lengths of corresponding sides. This is the scale factor, n.Then verify that nAB= DE, nBC =EF, and nCA= FD.

Type your response here:

Lengths of Corresponding Sides / n / Verify
DE ≈ / AB ≈ / nAB≈
EF ≈ / BC ≈ / nBC≈
DF = / CA = / nCA=
  1. In general, what do your observations suggest about the corresponding sides of two similar triangles?

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

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