## SIMILARITY OVERVIEW

As our next semester gets started, we get to turn our attention more directly toward the idea of trigonometry. Trigonometry is the study of triangles, their properties, and the relationships between them. Later this semester, this conversation will develop into a discussion of the functions sin, cos, tan, csc, sec, and cot. We’ll also eventually tackle the Law of Sines and Law of Cosines. But first, we need to look carefully at what makes triangles similar, why we care, and how to prove similarity relationships between figures.

Unit 7 is all about similarity and triangles. Some of the concepts will feel familiar, but we’ll take each of them further as we investigate the ins-and-outs of similarity in triangles. During this unit we’ll go through three major topics:

**Similarity**in shapes, dilations, and proving figures similar**Similarity in Triangles**including the proportional relationships that exist within all triangles, including right triangles**Build**a series of small models with the laser to illustrate the relationships between perimeter, area, and volume (weight) of similar figures.

When you’re done, you’ll have covered the first steps toward understanding trigonometric relationships!

**PART 1: SIMILARITY**

*(50 pts) About 3 days*

The first part of our unit is all about similarity between figures. We’ll look at the transformation *the dilation* and what happens to figures when they undergo dilation. These ideas will evolve into our next group of geometric proofs: *proofs for similarity!*

**PART 2: SIMILAR TRIANGLES**

*(60 pts) About 3 days*

The second part of our unit covers what similarity tells us about triangles. We’ll look at what happens when we dissect a triangle by subdividing segments. Here we’ll see proportions play a huge role in our understanding of similar triangles.

**PART 3: MODEL IT!**

*(140 pts) About 3 days*

Of course our unit will end with a review and test, but in addition you’ll get to build a model of similar figures! You’ll be asked to create a series of laser-cut similar triangles. You’ll ethen use those triangles to compare their perimeters, areas, and volumes (weights) to better understand proportions in similar figures!

# PART 1: Similarity

Dilations and similar geometric figuresThe first part of our unit is all about similarity between figures and what happens when a shape undergoes the dilation transformation. We’ll start by reviwing similarity and dilations. Last semester we looked carefully at different translations (rotations, reflections, etc); now we get to look at mathematical operations that scale figures to different sizes. You’ll start by taking a full page of notes on the four videos provided below. These videos cover the essential components of similarity, and will give us a good foundation to begin our serious work in trigonometry. In addition, you’ll work through three homework assignments on these topics before completing the part 1 *Quiz 17: Similarity*!

**GRADING & PROCESS**

Use your engineering notebook to take a full page of notes on similarity between transformed shapes, dilations, and proving similarity

Complete HW37: Dilations and Proof

Add to your notes about similarity as you watch the *Corresponding Parts* and *AA Similarity *presentations

Complete HW38: Corresponding Parts

Complete HW39: AA Similarity

Take *Quiz 17: Similarity!*

Dilations

Similarity Proofs

Corrsponding Parts

AA Similarity

**What’s Due** In *Similarity Part 1: Similarity*

- DIlations/Similarity Notes 1
- HW37
- Corresponding parts/AA Notes
- HW38
- HW39
- Quiz 17: Similarity

Here’s what’s due in Part 1: Similarity

- Take notes on dilations, transformations, and shape similarity
- Complete HW37: Dilations & Proof
- Next, add to your notes on similarity with information about corresponding parts and AA Similarity
- Complete HW38: Corresponding Parts
- Complete HW39: AA Similarity
- Take
*Quiz 17: Similarity!*

# PART 2: Similar Triangles

Investigating proportional relationships within & between similar trianglesThe second part of our unit is all about what happens when we divide triangles into smaller parts by subdividing opposing sides. In short: if we draw a line through a triangle that is parallel to one of the sides, it creates a smaller triangle that is similar to the original triangle. We’ll develop this idea more formally throughout the next few topics and assignments so that we can eventually turn these ideas into the six trigonometric ratios!

**GRADING & PROCESS**

Use your Engineering Notebook to take a full page of notes on the *Triangle Proportionality *and *Subdividing Triangles* presentations

Complete HW40: Triangle Proportionality

Take one more complete page of notes on the *Proportions in Triangles* and *Similarity in Right Triangles* presentations. Add examples from your homework to your notes on these topics!

Complete HW41: Subdividing & Proportions

Complete HW42: Right Triangle Similarity

Take *Quiz 18: Similar Triangles!*

Triangle Proportionality

Subdividing Triangles

Proportions in Triangles

Similarity in Right Triangles

**What’s Due** In Similarity* Part 2: Similar Triangles*

- Triangle Proportionality Notes
- HW40
- Proportions & Right Triangle Notes
- HW41
- HW42
- Quiz 18: Similar Triangles

Here’s what’s due in Part 2: Similar Triangles

- Take notes on triangle proportionality and subdividing triangles!
- Complete HW40: Triangle Proportionality
- Take one more page of notes on proportions in triangles and similarity in right triangles
- Complete HW41: Subdividing & Proportions
- Complete HW42: Right Triangle Similarity
- Take
*Quiz 18: Similar Triangles*

# PART 3: Model It!

Create a laser-cut series of modelsThe last part of our unit gives us the chance to build a series of laser-cut similar triangles and investigate their properties to confirm both the functions of Autodesk Inventor as well as the mathematical relationships between similar shapes! First you’ll need to plan out your 4-triangle series in your engineering notebook, and then build the proper triangles of various sizes within Autodesk Inventor. You’ll use the iProperties tab to record perimeter, area, and volume of each before exporting a .DXF file for use with the laser. After you’ve laser-cut your 4 triangles, you’ll measure their perimeters, areas, and weights to see how these numbers compare to the values calculated by Autodesk. You’ll also get to compare these values to see what they tell use mathematically about similar figures.

Of course our unit will also end with a unit review, a chance to get questions answered, and our final unit test to wrap things up!

**GRADING & PROCESS**

Use your Engineering Notebook to make a plan for creating your custom-cut laser models.

Use Autodesk Inventor to create your triangles each of the proper size. Use the iProperties menu to record the perimeter, area, and volume of each of your four triangle models.

Export .DXF files of your triangles and cut them on the laser out of acrylic. Measure and record the perimeter, area, and weight of each of your four triangle models.

Compare your measured values to those calculated by Autodesk, and to investigate the relationships between similar figures

Complete the Unit 7 Review

Schedule time to talk with Benshoof and get questions answered

Take *Unit Test 7: Similarity!*

**What’s Due** In *Similarity Part 3: Similar Laser Triangles!*

- Triangle Plan
- Autodesk & iProperties
- Laser Cut
- Compare Triangles
- Unit 7 Review
- Question Time
- Unit 7 Test

Here’s what’s due in Part 3: Build It!

- Make a plan for creating your custom-cut laser models.
- Use Autodesk Inventor & iProperties to create your triangles each of the proper size
- Laser cut them out of acrylic & measure properties
- Compare your measured values to Autodesk, and investigate the relationships
- Complete the Unit 7 Review
- Schedule time to talk with Benshoof and get questions answered
- Take
*Unit Test 7: Similarity!*