# Segment Similarity: Circles Part 4

So this happens in geometry:

It relates back to dilations.

The big idea: similar triangles have proportional measures.

## Scene Change

And then we see this:

## And formulas like these:

Wait, did you see that?  Down at the bottom with the big red arrow.  It’s fine print but it looks like a proportion of some sort.

## Notice and Wonder

Max at MathForum shares the awesome of the notice and wonder practice in a math classroom.  This is what I was experiencing when I saw (a)(b)=(c)(d) and some occasional references to a proportional measure of  .  I was noticing the proportions and wondering about  similar triangles, dilation and those proportional relationships.

So This was my first date with Geogebra.  I was hooked, and I never looked back.  Geogebra allowed me to visualize my intuition and confirm my conjecture.  Eventually I came up with an applet.

## Applets

It wasn’t enough to just show how flipping a triangle could reveal the proportional relationship with similar triangles.  We need to also build those proportions using actual measures.  I made it so that the user could drag text boxes that correlated to the needed measures from the circle.

## Where’s the similarity?

But wait, are all the relationships the same then?  Not exactly.  I would guess that this is also why most texts/reference guides give 3 separate theorems.  What each instance does have in common is there are two triangles, one similar to the other.  We just need to find the reflection that reveals the proportional relationships.

## Lo-Tech

I can imagine others thinking right now, “That’s cool and all, but I don’t have tech for each of my students,” or “I’m not really a techie person.”

So use paper. (Patty paper works great).  Or transparent stuff like sheet protectors.

## Paper Practice

Let’s be real.  This is great for conceptual understanding and visualizing the underlying relationships, but we need to practice.  And practice often works hand in hand with paper.  So we work with this concept for a half, maybe 3/4 of a period in the applets and then we practice fluency.  I actually go back to something like the Kutasoftware sheet from the beginning.  The difference this time is that we color code the segments, redraw them as embedded similar triangles, and use proportions.

And not once did we need those similar, yet not so similar theorems for circle segments.

# Clock Tangents: Circles Part 3b

Recently, there was a post looking at how to use the relationships of segments that are tangent to a circle to calculate more than just congruence.  Somewhere, and I’m not sure where, I remember Regan Galvan putting out there a tweet/post/comment about tan-tan angles on a clock.  Upon searching for it I found this:

I wanted to know how this thing worked.  So I built my own.

I purposely left out a lot of the information. I didn’t give any measures for angles or segments. This would be something that is developed in the classroom. The user and/or facilitator should be developing questions like:

• How do I measure the angle between the clock hands?
• How far away is the intersection point?
• Is the intersection point always outside the circle?
• What time is it when the two hands are perpendicular?
• What time is it when the two hands are opposite rays?
• Besides 12:00, what time is it when the two hands overlap?

Getting students to investigate these and asking to show evidence and create an argument to support their claim would be so much more fun than just calculating pre-made samples.  For students wanting more feedback and support, they can use the visnos interactive application.

# Satellites Spinning Round: Circles Part 3 (Tangents and Incircle)

## Inspiration

Have you ever wondered how far a satellite is from the planet?  Sometimes people think about these objects flying through the sky and rotating around a planet.  Such wonders have even inspired greats like Isaac Newton and Kepler.

Some good folk over at illustrative mathematics put this whole satellite into some simpler geometry with circles, triangles, and tangents.

Inspired by these I sought out to simplify the idea of tangents and how they fit around a circle.  We played with this:

## Questions

###### Warm up with some questions about the applet,
• What shapes do you see?
• Which measures, do you think are congruent?
###### go into some more typical math questions,
• What is the perimeter of the triangle?
###### get a little geeky
• What is the total area of the triangle (circle included)?
###### Then get all geeky advanced.
• How do you calculate the value for the area of the blue shaded area (circle excluded)?

## Student discourse:

Along the way here were some awesome teachable moments:

T: What kind of segments do you see?
S1: Tangents
T: How do you know?
S1: ’cause it says ‘tangent’.
T: I’m still not convinced. Maybe I mislabeled it. Prove it.
S2: They’re perpendicular. Tangents are perpendicular.
T: Anything else? Any other types of segments or is that it?
[crickets]
T: What about inside? Any segments inside the circle?
T: Where?
S3: Here?
T: Where? Pretend I’m blind.
S3: You already are blind Butler, that’s why you wear glasses.
T: [fake smile]
S3: Here at AJ. Ohh, and also AF. And these ones to but you didn’t label them.
T: Do you notice anything else? What if you drag around that red thing?
S3: They’re all the same?
T: What are the same?
S3: The radiussus. Does that mean they’re equal?
T: What do you think?
S3: Yeah, equal. They have to be, cause of the red thing.

S: Hey Butler, is that a kite?
T: It sure looks like it huh? What do we know about kites?
S: There’s stuff that’s the same. Like congruent.
T: For example…?
S: The pieces that match.
T: So how can you use that?
S: They’re equal. Oh wait, they’re equal. Sooo….
And then the student goes on to apply that understanding to fill in the blanks for the missing segment lengths.

The idea of using a few measures, congruent relationships, and some triangle calculations helped the students apply the geometry and calculations further than just one calculation.  As a follow up assessment they completed a static version of the question.  This was easy to create by just taking a screenshot of the applet at some particular balance.

Some day I want to advance this further with some clock math.