Geometry

GEO9 1) We learned that Gauss had determined which regular polygons are constructible with straightedge and compass. Use his result to make a list of which of the regular polygons with up to 60 sides are constructible For more info about this) - as we know this is more of a number theory question at this point. 1.1 Next, what is the largest known Fermat prime? 1.2 And, finally, what is the largest polygon with an odd number of sides that can be created 2) Please construct a heptadecagon using the instructions given in the history of the heptadecagon(17-gon) there and turn in your construction - if you do this with compass and straightedge then you'll get full credit for this one even if your heptadecagon is reasonably close to being regular (it's very hard to make it completely perfect!) Feel free to use GeoGebra or GSP to create it (then it should be pretty much "exact"!) 3) Please try your hand at the following construction - given a line and a point not on the line create a new line through the given point that's parallel to the first line (this shows up as Prop 31 in Euclid but please do this construction on your own). The goal for this exercise is to come up with the simplest way to do this, so count your steps and see how low you can go! Please be sure to justify your construction, i.e. explain why it is that you know the line you created is indeed parallel to the given line. In your justification you can

go ahead and add lines to your diagram if they're needed for your justification even if they weren't required in the actual construction. Count your steps in the usual way - every new circle and every new line constructed counts as a step (extending an existing line doesn't count as a new step, and picking a random point doesn't count either). Also go ahead and assume you've got a rigid compass to use (i.e. one that can transfer distances). 4) The Common Core State Standards emphasize work with parallelograms. By adding the squares to the sides of any quadrilateral and connecting their centers, one can create a new quadrilateral ((please bellow it’s the blue square).

a) Do some Internet search to make comments on this type of geometric construction (i.e the mathematics teacher 2016)

b) Construct the four angle bisectors of the angles formed by the intersection of the diagonals. 
 Create the intersection point of the angle bisector and the side it intersects to create the new quadrilateral. �

5) And now - please show why it's the case that using the rule that says that the new quadrilateral is always 
the same type as the original one—that is, they 
are self-duals. Further the new quadrilateral lengths are one-half those of the original, and the new quadrilateral is larger than the original by a (linear) factor of 2. One always gets back the same quadrilateral one started with. To argue this thoroughly one will need to consider similar triangles, use Prop 27 or 28 to make arguments about angles and parallel lines, etc. - good luck! Hint: You might want to use the Triangle Midpoint Theorem that states that if you connect the midpoints of any two sides of a triangle then that line segment will be parallel to the third side, and half the length of the third side. 6) Suppose that you've been given a 5 sided figure where you are told that all five sides are the same length. Now of

course that alone doesn't tell you much about the overall shape of the figure (try drawing out some figures with just this one constraint to convince yourself), however you're also told that 3 neighboring interior angles are all equal to each other. Now you're asked to prove that the figure must in fact be a regular pentagon (i.e. this means that you need to show that the last two interior angles must also equal the other three interior angles). The trick for this type of question will be to draw a fair number of triangles inside the figure and then show equality of as many angles as possible using vertical angle equivalences, summing of angles to 180 inside triangles, angle equivalences from isosceles triangles, etc. Good luck 7) Finally, an interior angle problem (along with a solution!) Consider a seven pointed star created as follows - start with seven points evenly spaced around a circle that are then connected to every other point as follows:

Now, figure out the sum of the seven interior angles in this star (i.e. the ones that are marked in the following picture). You should be able to use your knowledge of how to find interior angles in regular polygons along with a bit more triangle/angle sleuthing.

8) And now a solution to the last problem (so solve the last problem first, and then check your answer!) Please take a look at the following Proof by Picture concerning a generalization of the star figure from the last problem: Sum of Angles in a Star Polygon . To get credit for this problem, figure out what is going on with the picture proof and then write down the sum of the interior angles in a {5 / 2} star polygon.