Geometry.

1) Now it's time to finally finish off the Pentagon construction! Given that we know that we need to create Phi to make a pentagon, then if you start with a line segment one inch long then just using straightedge and compass show how one can produce a line segment that is (1 + Sqrt(5))/2 inches long (where Sqrt(5) means the Square Root of 5). To do this be sure to write down the steps you take (and you should also actually follow these steps to construct such a line segment). Try to find the construction that uses the fewest number of steps - again, please add enough written detail about your construction so it's clear what you've done. 2) And now, use your Phi construction from problem 1, and put it to work - if you're given a segment one inch long, then show how you can create a pentagon with that segment as one of the sides... think Golden Triangle - good luck! 3) A follow up on the curious properties of Phi. We found that Phi2 = Phi + 1, and that then Phi3 = Phi (Phi2) = Phi (Phi + 1) = Phi2 + Phi = (Phi + 1) + Phi = 2Phi + 1, and so on through higher powers of Phi, seeing consecutive terms of the Fibonacci sequence at work (for instance Phi4 = 3Phi + 2, Phi5 = 5Phi + 3, etc.) Now, go ahead and figure out what 1/Phi = Phi -1 , Phi -2, Phi -3 , Phi -4 equal in the same way as expressions just involving Phi and integers, i.e. figure out whether each of these negative powers can also be written as A Phi + B (where A and B might be negative integers now). 4) We have created equilateral triangles, squares, and pentagons at this point. Creating a hexagons is easy enough to do using six equilateral triangles. Believe it or not, there is no construction possible using just straightedge and

compass that creates a heptagon (the name for a regular seven sided polygon) . Now, try your hand instead at creating an octagon in as few steps as possible. Write down a summary of the steps you took to do your construction (try to make it as simple as possible) - note that you don't need to write down every single construction step, you can simply say things like "construct a perpendicular line to..." or "bisect the angle..." (the goal here would be to have a summary that's clear enough so that anyone else in our class could understand what you've done). 5) And again, believe it or not, there's no construction (with just straightedge and compass) that gives a regular nonagon (a regular nine-sided polygon). But it is possible to create a regular decagon (a 10 sided polygon). Give this a shot (i.e. create a regular decagon) - think back to your knowledge of how to construct a pentagon to help you out. Again write down a summary of the construction (again at the level of detail so that someone else could follow your construction). If you want to start with a "construct a pentagon as shown in a previous problem" then that's fine (i.e. you don't need to reconstruct a pentagon from scratch since you already did this). 6) Search the internet to find a few more examples of the Golden Rectangle in nature or art or architecture. Next find an example of the Golden Spiral in nature or art or architecture (this is the one that showed up going around the picture of the Mona Lisa). 7) Okay - give it a shot - supposedly the human body follows the Golden Ratio in many aspects. So go ahead and try to find as many instances of the Golden Ratio on your own

body as possible(!) For instance, supposedly the ratio of your overall height to the height of your belly button is equal to phi (hmm... really?!) - also consider ratios of the lengths of body parts between joints (for instance measure your fingers between the various joints, and/or your upper arm/lower arm, upper leg/lower leg, etc.) To help you out, feel free to take a look at https://www.goldennumber.net/human-body/ (Links to an external site.) Links to an external site. . For this question write down at least three examples that you're able to find - good luck! (note - your ratios might not be exactly equal to phi... but see if you can get them as close as possible!) 8) Now it's time for a return to the Pythagorean Theorem! First, finish off the proof based on the picture below. The right triangle in question is ACB, which is congruent to triangle DFE. As I mentioned in class, first calculate the area of triangle ADE (yes, it's missing an edge in the diagram!) by using 1/2 base DE times height AB. Next calculate the area of the same triangle ADE but now using base AE times height DF. The tricky part is to figure out the length of AE... it's equal to AC, which is one of the sides of the right triangle in question, plus CE. You can figure out the length of CE by a little similar triangle work, noticing that triangle BCE is in fact similar to triangle DFE (which is congruent to ACB) - good luck!

Question 8 proof

9) And finally, time for the Pythagorean Proof Party Contest! Please look to the slides bellow and look through the 14 proofs, they're all numbered with a number in the upper right hand corner of each of the 14 proofs. Also. Also look at Chris' amazing handmade wooden Pythagoren Proof Puzzle! Now select on favorite for each of the following categories (you can select for the same proof in more than one category if you'd like): Most Beautiful/Elegant Most Mathematically Interesting Most Far Out(!) Bonus question- Just for fun. Please take a look at the following Pentagon Construction Diagram . The construction is actually about creating a pentagon that is inscribed in a given circle - this is a different construction from creating a pentagon on a given base, which you worked on earlier in this problem set. For this bonus question (and this is not at all simple!) prove that the construction given here for inscribing a pentagon in a circle actually works (i.e. that the end result is in fact a perfect pentagon). To do this

you'll likely want to find a segment that has something to do with phi in this construction (look at segment DH) and/or you might want to show that it has the correct interior angles (e.g. 36, 72 or 108 angles), or some other approach entirely - good luck! Note that a few of the construction steps might seem just a bit cryptic at first - for instance "Circle AB" means draw a circle centered at A with radius equal to AB.