Tease your brainHow much do you know about circles? Does the Pythagorean theorem sound familiar? How about inscribed angles and their corresponding central angles? Know the area of a sector?
Here's a circle puzzle that was recently passed along to me by a friend. The task? Figure out the radius. Only two measurements are given (unless, of course, you count the right angles): the hypotenuse of a right triangle inscribed in one quadrant of the circle and the distance of one vertex from the circumference of the circle. Believe it or not, that's plenty of information.
So: what is the radius? The friend who sent me the puzzle said he saw “a horrendous variety of wrong solutions/answers/guesses.” Want to contribute to the variety?
Don't worry. I'm not giving the answer away all at once. Go read the comments. You'll probably find an answer there soon enough. And if you don't provide it, I'll post it myself.
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29 comments:
Am I allowed to answer here? I'm reluctant to give it away in the first comment.
Will you share this with your students?
So I guess the hint is to not think too hard, or perhaps not to think about it at all. :^p
Yes, anyone is allowed to answer here. Yes, I will share this with my students.
You may, however, want to hold back just a bit before revealing the answer. Maybe wait a day or two.
Perhaps the problem is in fact not a problem at all...
I see one very nice and simple answer and route there. Here provided in rot13, to hide it for those who want to think it through themselves:
Tvira gung gur gjb qvnzrgref qenja ernyyl ner qvnzrgref, gur guerr natyrf tvira ner rabhtu gb fubj gung gur gjb gevnatyrf fheebhaqvat gur tvira ulcbgrahfr npghnyyl sbez n erpgnatyr, jvgu gur tvira ulcbgrahfr nf n qvntbany.
Abj, gur gjb qvntbanyf bs n erpgnatyr unir gur fnzr yratgu. Urapr, gur nafjre jbhyq or gung gung yratgu bs gung ulcbgrahfr vf, ng gur fnzr gvzr, gur npghny yratgu bs gur enqvhf.
Abj, zber vagrerfgvat jbhyq or gb frr gung gung tvira cneg bs gur enqvhf bhgfvqr gur erpgnatyr vf rabhtu gb qrgrezvar gur natyr sebz gung enqvhf gb gur cbvag ba gur pvepyr gung zrrgf gur erpgnatyr. V qb abg, nynf, guvax gung vg obvyf qbja gb bar bs gur irel cerggl gevtbabzrgevp inyhrf, ohg V arire obgurerq ernyyl purpxvat vg guebhtu.
Answer Here
Don't think about it too hard. I spent about 20 minutes on it, trying three different strategies before I saw the answer. Damn, I love that problem.
Brilliant!
OK, I think I've got it - I was dimly aware of high-school geometry and thought about it a little while, and remembered what a square root was, and ALSO enlisted a calculator because I am math-challenged. I hope it wasn't simpler than that. Now what is Rot13?
but... but.... isn't the pythagorean theorem unnecessary -- and the 3" irrelevant?
ok.. i guess if you need to write a formal proof you need the pythagorean. but it's as plain as day.
For the poster asking about Rot13:
It's a shift cipher. Replace a with n, b with o, c with p, et cetera, and you get the original text. It's named "rot13" because, if we let a=1,b=2,c=3 and so on, each letter is replaced by the letter with a value equal to n+13 (mod 26).
I spent ten minutes writing out equations before I got it. It feels like I should have got it right away though.
Yes, the trick is not to think about it too hard. And certainly you shouldn't fill up sheets of paper with equations. The 3" is a lovely bit of misdirection.
I LOVE the Internet!
http://www.rot13.com/index.php
Am I missing something? Aren't the two diagonals of a rectangle equal? I must be out of it.
I think that a better phrase to use would have been " Believe it or not, that's more than enough information." But maybe that would give away the solution.
zeno, here's one for you!!
http://tinyurl.com/2kzynr
I've been doing a lot of geometry without a calculator lately - I'm taking an art welding class - so I've been thinking a lot about problems like this.
Still, I was surprised how quickly the answer came.
I'm stunned. I saw it right away. Then again, I'm not a mathematician so maybe I don't know enough to get suckered by the distractors.
The '3' is unnecesary, but only if the green line is an extension of the line above it. I know thats how it appears, but is it so ?
If its not Im in deep doo doo.
Oh well done. The first paragraph is a nice bit of misdirection.
I pretty much got it right off but I wonder if it's because I've not been doing math academically for so long that I did not approach this mathematically but as a brain teaser. I am curious if my experience was the norm here.
Thanks for the diversion. Beautifully tricky misdirection, it took far longer than it should have.
Neat one. I would have gotten it sooner, if I'd spent less time worrying about which trig identities I'd have to remember. These types of problems really bring out that anxiety in me.
For ignoramuses like me, can you post the answer now? I'm too dumb to understand mikael's decyphered rot13 post, assuming it's correct.
Okay, Tyro52: Here is the answer!
The radius of the circle is 8 inches. Notice that the given 8-inch line segment in the drawing is the diagonal of a rectangle embedded in the circle's quadrant. If you draw the other diagonal, you'll see it's the radius of the circle. The diagonals are necessarily equal, so the radius is 8 inches.
There's plenty of extraneous information. The business about the 3-inch line segment is purely a red herring. As noted by some commenters, it may have helped to be mathematically naive. That way you looked at the figure instead of immediately starting to compute distances or write down the Pythagorean theorem. Perhaps!
Right about now, I feel like uttering a hearty "D'oh!" For some reason, the obvious failed to dawn on me (this happens a lot, unfortunatley), which is that every line drawn from the center of a circle to its perimter is radial.
Thanks for lifting the curtain, or should I say removing the gauze from my eyes.
I figured it out in 0.47 seconds flat.
Diagonals of a rectangle are equal. That's all you need to know.
The problem is unsolvable since you've failed to label the center of the circle.
Don't be annoying, Adrian. The circle is divided into quadrants (or didn't you notice the use of the word?), which implies perpendicular diameters. Unless you are being deliberately perverse (or have reason to believe that the puzzle poser is), the sensible approach is to assume that the problem is more than a petty trick. You may as well complain that no one said the green line segment coincides with the diameter it appears to lie on.
I loved this problem! I didn't get it at first but after a minute or so I remembered stuff about 45 45 90 right triangles.
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