Interesting math problem that I saw on-line

[Charles Link]

TL;DR

You have a 4 sided polygon inscribed in a circle. Chord lengths are 2,3,4,5 in any order. Find the radius of the circle. It is also of interest to work the same thing for chords of length 2,7, 11, and 14.

Problem is contained in the summary above.

 

[Charles Link]

For a hint on this problem, opposite angles of the 4 sided polygon add to $\pi$ radians when inscribed in a circle.

Edit: I'm going to add to this, that if some people are stuck on it, that I didn't solve it instantaneously either. I was able to solve it though.

 

[sbrothy]

I was just about to protest that it sounded more like an ellipse, but on my first read I mistakenly had the polygon on the outside!

EDIT: and now I’m not even sure if that makes any sense at all….

 

[Charles Link]

It does make sense because if you have 4 sticks/chords, you can always find a circle that is too large or too small. There is only one circle that works.

I'm going to add another hint or two: Use the law of cosines twice for two opposite angles. The $L^2$ on the side opposite this angle is the same for both. Solve for $\cos{\theta}$.

 

[sbrothy]

Man, I’m gonna use pen and paper for this one!

 

[sbrothy]

And…. I don’t have a ruler. This is gonna be a mess.

 

[Charles Link]

@sbrothy Use the law of cosines to find one of the angles, and once you have that, you have 3 points to determine the circle. You put the center at $(h,k)$ with the one point in the middle as the origin, and you can find $(h,k)$ and thereby the radius.

 

[sbrothy]

I know I have enough clues by now. Something tells me the solution relates to Pythagoras. It’s simple geometry in fact. I have a bunch of excuses though. A laptop gone up in flames, literally. And I’ve just eaten. Homemade lasagne with lots of of cheddar. So I might have to pass today.

 

[Charles Link]

See https://www.physicsforums.com/threads/trigonometry-problem-of-interest.1082657/

The second case I gave comes from this problem, where when you compute it, you will find the diameter of the circle is the same as the largest chord.

 

[Charles Link]

We apparently must have a small number of viewers. I'm surprised nobody solved these two cases yet and posted the results.

 

[Charles Link]

I might as well give the (approximate) answers to both of these, so that anyone who likes doing artwork can draw them up The first turns out to have $r \approx 2.6$ and the second has $r=7$. It might make things easier to draw the first one in inches and the second one in centimeters. Then the first one has a diameter of about 5.25" (just under), and the second one would have a diameter of just over 5.5" if my arithmetic is correct.