Hey all. Here’s your weekend challenge question. The prize this weekend: any $5 album from the Amazon mp3 store. I can only give you the prize if I can get in touch with you (using your email or Disqus/Google/Yahoo/Twitter/Facebook account), so please don’t be completely anonymous if you want the prize.
(m + n + p + 180)(q – b – d – 2e – a – c + r) = 3x + y
Based on the figure and equation above, what is x in terms of y?
Good luck, and have a great weekend! I’ll post the answer Monday.
UPDATE: solution below the cut.
Nobody got this, although I did get a few emails with valiant attempts. The answer is: x = –y/3.
Why? Well, it’s a geometry plug-in. You’re given a bunch of shapes and not a single angle to call your own, so MAKE THEM UP. You just need to make sure you don’t break anything in the process — all your triangles need to add up to 180°, all your straight lines need to add up to 180° and in this case your pentagon needs to add up to 540° (because that’s how a pentagon rolls).
Easiest way to go is just to pretend your pentagon is regular (even though it’s clearly not) and plug in 108° for m, n, p, q, and r. That’s going to make all the base angles of the triangles come out to 72°, and a, b, c, d, and e all equal 36°.
The (m + n + p + 180) part of the question is horse-hockey. I only put it in there because I didn’t use those variables elsewhere in the problem, but I named them in the diagram.
The (q – b – d – 2e – a – c + r) part, using the numbers I just plugged in above (or any numbers that don’t break triangles, straight lines, or pentagons) comes out to ZERO. (Since nobody won the prize for this question, I’ll award it to anyone who feels like writing a general proof of this. I won’t hold my breath.)
So you’re left with:
Which is why x = –y/3.
|Come at me, bro. (Source.)|
38 degrees should be 36 degrees in explanation. Nice question…
=(q+r) – (a+b+c+d+e) – e
a+b+c+d+e = 180 (sorry, not going to name all the interior triangle angles to spell this out)
=(q+r) – 180 -e
e= 180 – (180-r) – (180-q) (obvious, right?)
so = (q+r) – 180 – (180 – 180 + r – 180 +q)
= q + r – 180 – 180 + 180 – r + 180 – q
all cancel out.
Yeah, I know, far from a proof, just presented for fun.
Yikes! Thanks JD. Fixed. That’s what I get for posting before coffee kicks in. 🙂
Very nice! Yeah, not a technical proof, but it’s sound. You corrected a typo of mine, too, so you’ve earned the prize. 🙂
Let me know which record you’d like, and I’ll gift it to your email address.
Thanks vm but please save it for one of the kids on here. I am just trying to sharpen up my skills to help my nephew who is taking the test soon (and my own kids in the years ahead). This stuff is tougher than I remember but I’m getting faster 🙂
Ah, gotcha. Don’t worry too much about this one, it would never appear on the test. Good for sharpening skills, though, as you said.