Off on a Tangent 
A Fortnightly Electronic
Newsletter from the Hope College Department of Mathematics 
October 28, 2016  Vol. 15, No. 4 
http://www.math.hope.edu/newsletter.html 
Upcoming Colloquia 

The following additional colloquium is on
the
schedule
for this semester, but more should be added.

Hope College Math Club Facebook Group 

If you use Facebook, you should join the Hope
College Math Club group. Recent posts include an animated graph showing
the increase in partisan politics in the US House, an article on active
learning, the mathematics found in diving, and 5 simple math problems
nobody can solve.
You might also see a picture of one of your classmates, or maybe even you! You can find the math club group page at https://www.facebook.com/groups/HopeCollegeMathClub/. 
Math
in the News: Can playing video games improve math and science ability? 
According to Dr. Alberto Posso,
from RMIT University in Melbourne, Australia, teenagers who regularly
play online video games have higher test results in mathematics and
science. He used
the test results from more than 12,000
Australian 15yearolds in mathematics, reading and science, as well
survey results on the students' online activities. Posso said, “Students who play online games almost every day score 15 points above the average in maths and 17 points above the average in science. When you play online games you’re solving puzzles to move to the next level and that involves using some of the general knowledge and skills in maths, reading and science that you’ve been taught during the day.” He also went on to say, that students who used Facebook or chat everyday scored 20 points less on average, than students that never use social media on the mathematics portion of the test. (I'm sure this does not include those that look at the Hope College Math Club group Facebook page.) The research, "Internet usage and educational outcomes among 15yearold Australian students" has been published in the International Journal of Communication. 
Problem
Solvers of the Fortnight 

In our last Problem of the Fortnight we
presented the following conversation between Math Man and Vectoria.
"Hey, Math Man. What are you looking at?" "Oh, hi,
Vectoria. I'm trying to figure out this geometry problem."
"Oooh! What is it? I love geometry!" "Well, we have a quadrilateral ABCD, and ABC and ACD are similar right triangles with right angles at B and C respectively." "Okay, I'm following. Go on." "This point E here is the midpoint of BC." "Yeah, okay. . . ." "Well, the problem is to show that BD and AE are perpendicular." "Hmmm . . . . That is an interesting problem. Let me think . . . ." After a while, Vectoria exclaims, "Got it! Wow, that is a cool problem, M2  and you can solve it using vectors! In case you didn't know this already, I like vectors a lot!" "I hadn't made that connection before, but now that you mention it, it makes perfect sense. . . . Or wait, I would have thought that your parents were the ones who really like vectors, since they named you Vectoria." "Well, they do, but I like them a lot too. Our family was always headed in pretty much the same direction." "Oh, I see. . . . But wait, do you need to use vectors to solve this problem?" "No, I wouldn't think so. But if you do know a little bit about vectors, the solution's quite elegant." After a brief pause, Vectoria says, "Well, see you later, Math Man. I need to head out now." "Wait, can't you stay to tell me the solution?" "Nope, sorry, M2. Gotta run." "Well, can you stay long enough for me to take a selfie with you? I mean, everyone's going to want to know what you look like! And besides, I kinda don't like being the only one in these photos." "Nope, sorry again, friend. I really do need to run. Maybe next time, okay?" A little dejected, Math Man says, "Okay, yeah, maybe next time. . . . See ya!" On her way down the hall, Vectoria turns and says, "Yeah, see ya later, Math Man. Have fun working on the problem! It's a good one!" At that moment, Math Man pulled out his phone to try to get a picture. However, either Vectoria was too fast or Math Man was too slow, so all he got was the picture shown. Hopefully Vectoria will slow down enough next time so Math Man can get that selfie with her. The Problem this fortnight is to help Math Man figure out how to show BD and AE are perpendicular. Congratulations to Richard Edwards, Jiyi Jiang, Philip LaPorte, Zheng Qu, and Sam VanderArk  all of whom correctly solved the Problem of the Fortnight in the last issue of America's premiere fortnightly electronic mathematics department newsletter. 
Problem
of the Fortnight 
"Hey,
M's. What'cha doin'?"
"Oh, hi, Vectoria." Math Man hands Vectoria the piece of paper he was looking at and says, " I'm trying to figure out what this is." After a few moments, Vectoria says, "Oh, I get it! It's a proof!" "Really? A proof of what?" "It's a trig identity involving an inverse trig function." "Which one?" "Well, it could be any one, I suppose." "No kidding. . . . I still don't get it. Which triangles do you use to form the identity? "All three of them." "Hmmm. . . . This is really hard." "No it's not! It's as easy as. . . ." Glancing at her watch, Vectoria stops in midsentence as she realizes that she's late. "Whoa! Sorry, M's. I gotta run. I'm late!" "Wait, wait. . . . easy as what? Can't you give me a clue?" As she sprints straight down the hall, Vectoria says, "I think I just did! Maybe even two!" Vexed, Math Man sighs and stares at the figure again. "I still don't get it. I think I'm more confused now than before she came." The Problem of the Fortnight is to help Math Man discover the trig identities lurking in the figure. You don't have to list every conceivable trig identity that this figure contains, but giving three trig identities ought to help Math Math figure out what this figure is communicating. Write out three trig identities that use all three triangles contained in this figure and explain your reasoning carefully so that you can give Math Man the best chance of understanding what Vectoria saw in this figure. Write your solutions on a triangular piece of paper and drop it in the Problem of the Fortnight slot outside Professor Mark Pearson's office (VWF 212) by 3:00 p.m. on Friday, November 4. As always, be sure to include your name, as well as the name(s) of your math professor(s)  e.g. Tryg E. QuayShon, Professors Wright and Tryangoll  on your solution. Good luck and have fun! 