Download Takehome final questons

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.