Ok, I think it’s going to happen. This week I got through just over half the unit Circles which was further than my goal for this week (Woo!). I’m on track to get through the unit and finish the course High School Geometry by the end of next week, Week 26. Although I actually surpassed my goal for this weeks, I struggled with much of the content this week. I think I likely spent somewhere between 8-10 hours working on K.A. and had some serious moments of frustration followed by a few moments of clarity and some breakthroughs.

I began the week working through questions where I was asked to find the Arc Length and Arc Measures of circles when given other values of the circle. In order to solve most of the questions, I used the formula Arc Length/Circumference = Theta Degrees/360 degrees. What made these questions extra difficult for me was when I’d be asked to use that formula but replace degrees with radians. Even now, I’m still having a hard time wrapping my head around radians but have got a better grasp on them and managed to work through the questions and come to the correct answer. What I understand about radians at this point includes:

• Radians
  • A unit of measurement that can be used to measure an angle (a.k.a. the Arc Measure, i.e. it can replace degrees) and also be used in place of Arc Length.
  • 1 radian equals the distance of the radius. When you create a central angle from a circle that makes an arc length the same distance as the radius, that arc length AND the arc measure are both considered to be 1 radian in measurement.

Looking through my notes and trying to put the definition of radians into my own words, it’s clear that I still have a weak understanding of how they work and what they are. There’s no doubt that I have a better understanding than I did last week, but I clearly need to keep learning about them to feel comfortable using them.

I was then introduced to inscribed angles. An inscribed angle is an a circle created when two secant lines (i.e. lines that cross through a circle) share one similar point on the circumference and create an angle inside the circle from that point, a.k.a the inscribed angle. I learned that the arc measure of an inscribed angle is always half the value of the arc measure of a central angle if both angles share the same non-vertex points. I also learned that the arc length of an inscribed angle (i.e. the section of the circumference between the two non-vertex points) is half the value of the arc measure (i.e. the angle created at the vertex point). Again, this is something I still haven’t quite wrapped my head around and I find difficult to put into words. (As a side note, I once heard that if you’re unable to explain a concept in simple terms, you don’t have a great understanding of the concept. This is exactly how I feel right now.)

I was then introduced to inscribed quadrilaterals. These are four sided shapes inside a circle where all four vertexes sit somewhere on the circumference of the circle. I learned that the angles of the opposite corners of inscribed quadrilaterals always have supplementary angles, i.e. their angles added together equal 180 degrees. There was a very interesting “proof” video that explained why this is the case but, again, it’s another concept that would be difficult for me to put into words.

The last things I worked through this week were questions on tangent lines and circumscribed angles. As mentioned in my last post, a tangent line is a line which intersects a circle at only one point (imagine a ball sitting on the ground and the ground being considered the tangent line). I was shown proofs and learned why a tangent line is always perpendicular to the central point of a circle. Circumscribed angles are angles created when a point outside a circle is connected to the circle via two tangent lines. Another way to phrase this would be ” two lines are tangent to the opposite sides of a circle and the point at which they cross each other is considered the circumscribed angle.

Finishing this week, I feel somewhat conflicted. I’m happy that I’ll likely get through this course by the end of next week which was my goal, but I don’t feel confident in my understanding of the material in this unit which bothers me. I feel like my understanding of this material is improving but I definitely have a ways to go before I could explain all these concepts in an easy to understand manner. Right now I’m sitting at 960/1700 M.P. completed of this unit, Circles. Of course I’d prefer to know the subject matter inside and out before moving ahead but at this point I’m running out of time to hit my goal and am prioritizing grinding this unit out, getting 100% on the unit test, and getting through the course test by the end of next week. As I mentioned in a post awhile back, I HATE not hitting my goals so at this point my main objective is to get through the course by the end of next week no matter what. I don’t feel great about rushing through this unit though…