Apply transformations to prove that all circles are similar.

General Information

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Geometric Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Circle
- Dilation
- Similarity
- Translation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students learned about similarity and similarity transformations. In Geometry, students apply transformations to prove that all circles are similar.- Instruction includes presenting students with a pair of circles of different size and asking them to identify a sequence of transformations that would map one onto the other. Students should realize that a single translation and a single dilation is all that is needed in a sequence to map one onto the other.
- Instruction includes the connection to the coordinate plane by showing that two circles are similar using coordinates.
- Students should connect the definition of similarity in terms of corresponding parts
applied to polygons and explore what parts of the circles will be in proportion between
the preimage and image of a dilations.
- For example, given two circles, their radii ($r$
_{1}and $r$_{2}) and their diameters ($d$_{1}and $d$_{2}) would satisfy the proportional relationship .

- For example, given two circles, their radii ($r$

### Common Misconceptions or Errors

- Students may think that always need a formal proof to prove that all circles are similar.

### Instructional Tasks

*Instructional Task 1 (MTR.4.1)*

- Two concentric circles with point
*A*as the center and circle*B*are given on the coordinate plane.- Part A. Describe the transformation(s) needed to map the smaller circle
*A*onto the larger circle*A*. - Part B. List the transformation(s) that could be used to show that each circle
*A*is similar to circle*D*. Compare your transformations with a partner. - Part C. What is the difference in the transformation(s) depending on the circle
*A*chosen?

- Part A. Describe the transformation(s) needed to map the smaller circle

### Instructional Items

*Instructional Item 1*

- Circle
*A*and circle*D*are given below.- Part A. Describe a set of transformations that could be used on circle
*A*to show it is similar to circle*D*. - Part B. Describe a set of transformations that could be used on circle
*D*to show it is similar to circle*A*.

- Part A. Describe a set of transformations that could be used on circle

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

This benchmark is part of these courses.

1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

## Lesson Plans

## Perspectives Video: Professional/Enthusiast

## MFAS Formative Assessments

All Circles Are Similar:

Students are given two circles with different radius lengths and are asked to prove that the circles are similar.

Similar Circles:

Students are given two circles with different radii and are asked to prove that the circles are similar.

## Student Resources

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## Parent Resources

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