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MTH 250 takehome final questions Not a whole lot here – most of the second half was about knowing key results, and a lot of computations. It wasn’t as proof-y. So I had to dig a bit to find a handful of proofs questions, and they’re coming from the earlier parts of the material. Problem 1: Quickie warmup proof, just to get you back in the habit of proving things. Perpendiculars AE and CF are dropped from vertices to the diagonal BD of parallelogram ABCD . Prove that AE CF . Note that we’re in (Kay’s version of) Euclidean geometry at this point in the text. [Kay, Section 4.2, problem 16, page 235] Problem 2: OK, here’s a couple of key things I don’t think we proved! The proofs of the Two-Chord Theorem and Secant-Tangent Theorem were left as exercises for the reader in the text, and I don’t believe I proved them in the notes either. (If I did, don’t tell me – just enjoy it as a freebie!) Here’s your chance Problem 2 is the Two-Chord Theorem, and Problem 3 is the Secant-Tangent Theorem. Prove the Two-Chord Theorem by first showing that PAC and PBD in the figure have two pairs of congruent angles, and hence are similar triangles with proportional side lengths. Then complete the proof that PA PB PC PD . [Kay, Section 4.5, problem 18, page 281] Use betweenness, opposite rays as apparent in diagram. If you’re looking at the figure in Kay, do NOT treat the angles he’s marked as congruent as givens. Those are the angles that you need to prove are congruent to get your similar triangles – the goal here is to justify why 1 2 and 3 4 , and take it from there… Problem 3: In the figure below, secant PB and tangent PC are drawn, forming triangles PBC and PCA . Use similar triangles to prove the Secant-Tangent Theorem: PC 2 PA PB . [Kay, Section 4.5, problem 20, page 282] Use betweenness, opposite rays, as apparent in diagram. PC is tangent to the circle. Again, if looking in Kay, the marked congruent angles aren’t givens, they’re what you need to justify to move along with the proof. Hint: be sure you get the correct similarity; triangles ABC and ACP are NOT similar to each other. You’re aiming for CBP ~ ACP And that’s it. I’ve been scouring your textbook (and my brain) trying to come up with something that isn’t just duplicating a computation you’ve already performed on an assignment – I suppose I could have you go into Non-Euclid and construct and measure yet another figure or work out another hyperbolic line equation, but that just seems redundant. Most of the computations are of the sort that can be readily performed on an in class test, and we really are running out of things to prove! The bulk of the points for the final exam will come from the in class portion.