Geometry

1) Time to explore some more about hypercubes! For this problem please create a table that lists the number of component parts (vertices, edges, faces, cubes, and also including rows for tesseracts, and for a few higher order hypercubes as well) in "cubes" from various dimensions: a zero dimensional cube (just a point), one dimensional (a line segment), two dimensions (a square), three dimensions (a cube), four dimensions, called a "hypercube," or "tesseract." At this point, however, keep going with your table, up through seven dimensional hypercubes(!) - i.e. add three more columns covering five, six, and seven dimensional hypercubes respectively. Note that after four dimensional hypercubes, all the higher ones are also called "hypercubes," just differentiated by saying "five dimensional hypercubes" "six dimensional hypercubes" etc. There's an interesting website where you can see versions of the hypercube (and link to some other neat sites as well) - http://sayitwithscience.tumblr.com/post/11361440387/hyperc ubes-what-is-a-hypercube-also-referred-to Links to an external site. And here are the links to the hypercube videos we saw in class: http://giphy.com/gifs/animation-math-mathematics- wXnFJWlvfmtsA Links to an external site. and https://giphy.com/gifs/processing-hypercube-phxXlMiXA1Xjy (Links to an external site.) Links to an external site.

2) On to RASS! (Right Angle-Side-Side). you saw that it's not possible to come up with a congruence theorem for simply SSA, but if the angle is a Right Angle, then in fact RASS does hold For this problem I'd like you to go ahead and write up a complete proof, with everything (including justifications for each step) spelled out clearly. To begin with, we supposed that there were two triangles ABC, and DEF as shown below: Suppose you know that both have right angles (at vertices A and D respectively), and that AB = DE, and BC = EF. Now write out a complete proof that the two triangles are congruent. For your proof, please use the result of Euclid propositions 1 through 34. probably the simplest way to begin the proof is to start with triangle ABC, and then explain how to construct a triangle to its left, sharing side AB, as in the diagram below, in such a way so that you can show that the second (blue) triangle is congruent to triangle DEF. Next, show the blue triangle is congruent to the red triangle... On the other hand, feel free to try approaching this proof in a completely different way if you'd like! 3) Another proof for you using triangles! Prove that the diagonals of a rhombus are perpendicular bisectors for each other (hint, it's all about triangle power - use your triangle congruences!). Recall that the definition of a rhombus is in fact given in Euclid - use this definition to work from. At this point feel free to use any results up through Proposition 34 in Book One for your proof. 4) And one last rusty compass problem! Recall that you accidentally left your compass out in the rain and that it rusted in place so that you can only draw circles with radius

about 1 inch. Now, using a line segment that's around 4 inches long (but not exactly - it might be a bit more or a bit less than 4 inches, so you can't count on knowing its exact length in your construction), create an equilateral triangle with this given line segment as its base. Try to make your construction as simple as possible! As usual, in your answer please be sure to describe your construction so someone else could do the same thing. For this question, please go ahead and prove that your triangle is in fact an equilateral triangle (this should be pretty straightforward to do based on your construction). 5) Next go ahead to show and prove that if you build squares on the sides of a parallelogram and connect their centers, then that will produce a square. He more or less touched on all the aspects you'll need to go over for this proof - the last issue he looked at was trying to show all four triangles were congruent with some SAS congruences - good luck! 6) Last year, a student sent a link to a neat YouTube video of This Old House (a home repair show that started in Boston) showing off a clever trick for bisecting an angle going around the corner of a wall so that you can cut two pieces of wood trim that will then fit together perfectly. For this problem prove that this trick does in fact work using just results from Euclid’s Props 1 through 34. Thus in the picture below mimicking the results of what's done in the video, show that the angle marked at A (greater than 180 degrees), is in fact bisected by the dotted line connecting points A and B. Good luck! You'll probably want to draw a few more triangles to help out - you might want to create a

couple of right triangles, for instance, by drawing perpendicular lines from the point marked H (for "Hint!") to the opposite sides of each piece of wood trim - and then you'll also need some angle observations from Prop 29, along with few triangle congruence theorems as well! Here's a copy of the picture below in case you have trouble reading the captions on it: Overlapping Trim Board Angle Bisecting Trick 7) Following along the lines with the perpendicular bisectors of a triangle (that they meet in a point), now prove that the three angle bisectors of a triangle ABC meet in a point. To start your proof you need to first consider the intersection point, D, of two of the angle bisectors and then work your way around to showing that the line connecting D to the third angle does in fact bisect that angle. In addition to using any of Euclid's results up through Prop. 34, you might also need to use the RASS theorem for triangle congruence at some point. When writing up your proof be sure to write down each step you took in the proof, along with an explanation of why each of the steps holds (i.e. if you've determined that two triangles are congruent be sure to explain why this is the case). Finally, when you're finished with your proof explain what's special about this new "center" of a triangle (it's typically called the "incenter" - why?) 8) Next find out more about triangle centers by going to http://faculty.evansville.edu/ck6/tcenters/ (Links to an external site.) Links to an external site. First, write down the four triangle centers known to the Greeks listing how each one is defined and what makes it special (you can leave "orthocenter" blank for what makes it

special, or try to find something special about it online). Next use the http://faculty.evansville.edu/ck6/tcenters/ (Links to an external site.) Links to an external site. website to find at least one other "interesting" (by your definition!) triangle center, write down how to find it and what makes it special.