Circles all Lined Up

All right, we've looked at the rules for circles, and there are lots of them. It doesn't ever seem to be more than cutting a number in half mathematically, but there sure are a lot of things to remember. Let's see if you can create one yourself. Using some circle as a base (like a can or something), put in to it every rule we've learned in circles such that all the angles and arcs are appropriately marked and measured. Inscribed angles, tangents intersecting with diameters, congruent chords cutting off equivalent arcs, central angles, etc...
Your task is then to turn in to solvable problems, where enough information is given for another student to then solve for missing arcs and angles. There should be a minimum of 10 angles/arcs to solve for. An example is posted below this text. If you're not sure on how to proceed after looking at my example, I'll be glad to show you another physical copy in the classroom, just ask. This is going to be a high value HW assignment, so choosing to skip this one will be harmful to your overall grade. I appreciate your efforts!

Completion of this HW assignment is three-fold:
1) Draw the circle, put in the appropriate segments/lines and figure all the angle/arc measures for your self.
2) Take out a lot of the measures, leaving just enough for another student to be able to solve it themselves as best they can.
3) Actually give that drawing for the other student to solve (trade, if you will) and see if indeed if it is solvable. Remember, there is a minimum of 10 'answers' for the partner to try to get.
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Sin/Cos/Tan AKA: Trigonometric Ratios

Right triangles really are everywhere, the importance of them in building design cannot be overstated, and you also can find them on your trip into the mountains (and out of) on some road signs! Ever seen the one that states a warning to truckers that the 'grade' is 6%? This one:

What is this 'grade'? Sometimes these signs are accompanied with an actual percent, as this one is:

Okay, it says 7%. Percent of what? Well, when you travel through the mountains, you often see signs that say things like "Trucks check brakes -- 10% grade" or "6% grade -- Trucks use right lane only." These numbers obviously have something to do with the steepness of the road, but their exact meaning is a mystery to most drivers. 
The grade of something is simply a measure of its rise over its run. To understand rise and run, it helps to think of the hill as a big right triangle (a triangle with a 90-degree angle), like this: 

The rise is the length of side B, or the height of the hill. The run is the length of side A, the horizontal measure of the hill at ground level. So, if you rose 100 feet over a horizontal distance of 1,000 feet, rise over run would equal 100 divided by 1,000, or 0.1. To get the percent grade, you simply multiply by 100, which gives you 10%. It doesn't matter whether you use feet, meters, miles or kilometers -- if you know how far the road rises in a given horizontal distance, you can calculate the percent grade. 
In common practice, people often refer to percent grade as the rise divided by the distance you would travel going up the hill (side C), rather than the horizontal distance (side A). If you have an odometer and an altimeter, this is pretty easy to calculate. You check the altitude at the starting point and reset the odometer trip meter. You climb a certain distance and divide the change in altitude by the miles you've traveled. This is not technically the grade, but for normal roads that aren't very steep, it ends up being pretty close because the horizontal distance and the length of the actual road are nearly the same. 
To calculate percent grade exactly, you need to figure out the horizontal distance traveled (A). Since you know B and C, you can calculate A using the Pythagorean theorem -- "the square of the hypotenuse is equal to the sum of the squares of the other two sides," or: 
C2 = A2+ B2 
This means that: 
A = SquareRoot (C2 - B2) 
If you had driven 1,000 miles down the road and risen 100 miles, the horizontal distance would be the square root of 990,000, which is approximately 994.99 (rounded up). 
So what good does all of this do you? First of all, the percent grade gives you a relative sense of how steep the hill is. If you've climbed a hill designated as having a 5% grade, for example, you'll know approximately what to expect from any other 5% grade hill. 
. Percent grade also gives you enough information to figure out the grade angle -- the angle of the hill's ascent (angle d in the diagram below). If you know trigonometry, you've probably recognized that rise divided by run is equal to the tangent of angle d. To calculate the angle of ascent when you know the percent grade, you simply take the arc tangent of the grade (the inverse of the tangent). So, if you have a 10% grade, you look up the arc tangent of 0.1 and find that the angle is 5.71 degrees. 

Roads are measured in percent grade rather than degrees for two reasons: 
· You don't have to have a special calculator or trig tables to calculate percent grade. 
· If you know the percent grade, it's easy to calculate the approximate distance you have risen or fallen simply by looking at your odometer. If you have coasted down a mile-long hill on a 10% grade, you know you have fallen about a tenth of a mile in vertical height.

So, what would I like you to try? I would like you to construct a 'hill' that has approximately a 10% grade. Lots of choices here, you could use our new fallen snow(!), maybe Play-doh, cardboard and tape, dozens of books, or whatever you can think of. Ultimately, when you've completed this, I'd like you to either literally bring it intact into the classroom (which seems to me to be somewhat unlikely, but if you can...) or take an image of it and bring that in. Obviously, I'll discuss this in class a bit with you on Monday to ensure we all are on the same page, and I will gladly give advice/hints as to its completion, so don't panic. And I understand the explanation I just gave is slightly confusing, so I'll also make that as clear as possible. But hopefully you'll find this moderately entertaining, and have fun constructing!!  [link]