Tag Archives: proportion

Segment Similarity: Circles Part 4

So this happens in geometry:

Proportional Triangles: http://goo.gl/RDkMQo

It relates back to dilations.

Transformations Applet: http://ggbtu.be/m125888

The big idea: similar triangles have proportional measures.

Scene Change

And then we see this:

Segment Measures in Circles

And formulas like these:

Circle Segment Theorem 1
Circle Segment Theorems 2 and 3

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

Flipping the Triangle: APPLET

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.

Circles Part 1: Similarity Intuition

All circles are similar, right?

Okay, maybe it’s not given.  In fact, it needs to be proven.  This proof is yet another that is so easily demonstrated with dynamic math tools (like desmos and geogebra

Circle Similarity with Desmos
https://www.desmos.com/calculator/vscq2qtp2w

So long as you can move one center onto the other (translate) and dilate one radius to equal the other, similarity is achieved.  This works for every circle.  The perfect proportional balance achieved with circles lays the foundation for most of not all relationships found in them.

Proportional Measures

Similarity gives us a simple system for comparing measures in multiple figures.

2014-07-07_1512

You can even explore this with repetition of congruent triangles:

Dimensional Analysis with Scale
http://ggbtu.be/m127379

Applications

Below are a couple of applications that push further with exploring the measures of circles, proportionality, and relationships in the measures.

Pizza Pi 

This activity was first inspired by a posting on John Stevens‘ website wyrmath.wordpress.com .  It looks at proportions with a different angle (pun intended) and asks user to compare various portions of different sized (but similarly shaped) pizza.

Rolling Tires

Another great one from Andrew Stadel.  This is a 3 act math lesson looking at relationships with circumference of a tire and cumulative rotations.

Coming up next…

So where do we go from here.  Following the basic intuition of similarity and proportions in a circle, we can build into

  1. More Angle Relationships with circles
    • Central angles and/or non-central angles
    • Circles circumscribed around a triangle start to simplify some of these relationships.
  2. Segment Relationships
    • Tangents and Incircle
    • Dynamic Relationship with Intersection of Chords, Tangent-Secant, and Secant-Secant (hint: they’re all a similar relationship)

See you in Circles Part 2: Angle Relationships.