Geometry

1) First, going over a number of results we worked on in our last class, but from another, quite entertaining! perspective - please read Jim Tanton's Curriculum Essay on Circle Theorems. Jim is a great math writer/educator who leads workshops around the world - he's taught here at Harvard in the past as part of the Math for Teaching program (he's on our program's advisory board). For more about Jim, please see his website at http://www.jamestanton.com/ (Links to an external site.) Links to an external site. Links to an external site. 1 )Tanton's Curriculum Essay on Circle Theorems. For this problem please read the essay, and then do the exercise on page 5 (in the boxed out section, above Thales Theorem, on the right hand side) –hint ! you'll need to do some angle chasing . Also please note that you might end up showing the Cyclic Quadrilaterals result at the top of page 6 as well! 2) Here's another construction challenge that plays off of our knowledge about triangles inscribed in circles when one side of the triangle is the circle's diameter. At least in this problem you get to use your good old regular compass and straightedge - no rusty or broken tools! Given a circle with center O, and given a point A outside of the circle, construct a line through A that is tangent to the circle. Note, to be a classical Euclidean construction, you can't just slide the straightedge back and forth until it seems to be tangent to the circle - that's not a legitimate construction - you need to construct a second point on the desired tangent line before drawing the tangent as two points are necessary to determine a line. In your write-up be sure to write down the

steps you took for this construction (it probably won't be obvious from your diagram alone!) As a pretty big hint to get you started - consider the triangle connecting O, A and the point where the tangent line you're trying to create will touch the circle. These three points will be vertices of a right triangle as tangent lines are perpendicular to diameters of circles (you read about this in Jim's essay in problem 1). Now, we know that right angles fit neatly into circles This problem is a bit involved to figure out, but you should be able to do this in fewer than 10 steps. Good luck! 3) As one final tangent problem - now go the other way. Suppose you're given a line L with a point B on the line and another point A, that is not on the line. Construct a circle that goes through A and B and that is tangent to the line L. This is a bit tricky, but again you should be able to do it in fewer than 10 steps. As a hint, consider starting with a line through B that's perpendicular to L - the center of the circle you're trying to find should lie on this new line. 4) Here's a relatively simple proof question involving a couple of concepts. Let AB and AC be two tangent lines from a point A outside a given circle as shown below:

The picture is also here in case you have trouble seeing it. Show that in fact AB is congruent to AC (i.e. the two line segments have equal length). For this problem, just use triangle congruence theorems, not the Pythagorean Theorem, to prove this. 5) Here's a somewhat odd sounding question based on a practical matter in sports that picks up on one of the themes in Jim's essay involving chords and angle equivalences. Suppose you're playing soccer - you're going to try to score a goal by kicking the soccer ball through the goal. At the moment, you're standing 40 feet away from the goal, directly in front of the goal. The goal itself is 24 feet wide (please see soccer position diagram ). Draw a diagram of this situation for yourself (looking straight down with you marked as a point and the goal opening marked as a line segment). Now using your knowledge of geometry determine all the other points around the goal where if you're standing at any of those points, the angle of the opening of the goal (the line segment marking the goal) is the same as the angle was at the first point directly in front of the goal (and of course we talked about this in class). On your diagram determine these points with a simple straightedge and compass construction (and write down a summary of what you did and why it works to locate these points). In essence, this is a very longwinded way of asking for a pretty straightforward construction, but it does lead to some interesting real life applications of this particular result about chords! If you or one of your students likes (American) football instead, then this is something of interest to field goal kickers as well.

6) At one point a student in class showed me a truly excellent way to solve the tricky "rusty compass" problem (number 3 in set 4) - it involves several things we've worked on in the last several classes, so I thought it would be a neat way to revisit some topics. I'll coach you through the solution (which I'd suggested could take 12 or more steps). If you follow this through, you'll find an amazing six step solution! So, first please re-read problem 3 set 4 to recall the set-up. Now, the first step is to draw a circle centered on the line, as close to "directly below" point A as possible. Call the two intersections of the circle and the line points B and C. Next draw two lines - one connecting A to B and one connecting A to C. Step back - you've got a triangle, A, B, and C. In some sense you're now trying to find the altitude going through A and triangle side BC (that will be a line through A perpendicular to the given original line). We know that all three altitudes intersect in the so-called "orthocenter" point. Given our Thales Theorem result from class (Thales Theorem - please see http://mathworld.wolfram.com/ThalesTheorem.html) (Links to an external site.) Links to an external site. can you now find the orthocenter with just two more strategically chosen lines?! and then the line you want through A is ready to go! For this problem please include a diagram to show off that you've figured the problem out (and feel free to compare this approach to the one you'd found last week!) 7) And, finally, somewhat just for fun - following up on the phrase I've used a number of times (for better or worse!) - "let's start with a random triangle..." - here's a letter that was

posted in the journal Mathematics Teacher from October 2010 that addresses exactly this issue - Drawing a Random Triangle . Please read through Adam Kalman's explanation of what he feels is a useful triangle to have on hand for such drawings, and then to show you've gone through his letter, please draw such a "random triangle" with the angles he's suggesting using. What do you think of his choice?