Finding the Length of a Chord

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Carlington
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Finding the Length of a Chord

Postby Carlington » Sun Apr 10, 2016 9:58 am UTC

The particulars of how I reached this problem aren't super important.
Suppose you have an arc segment of known radius but unknown arc length. From the ends of this arc, two lines are constructed, perpendicular to one another, such that one line is vertical and the other horizontal. With this information, I want to find the length of the segment of the horizontal line that reaches from the end of the arc to the intersection with the vertical line. I've included the best attempt at a diagram I could put together on my phone below (I have no idea how big it will be):
Spoiler:
2016-04-10 19.37.31.png


In the diagram, for clarity, I want to find the length BD. I also have that BC:BD::2:3
(A is the centre of the circle)

My working so far has been to start by constructing a line CD to form the right triangle BCD. Since I know the ratios of the shorter sides, I can work out the other two angles in the triangle, which came out to be roughly 34 and 56 degrees. If I knew that BD passed through the centre, I would be able to subtract the arc length subtended by an angle of 34 degrees from the arc of half the circle to get the arc length CD. I could then use that to find the angle CAD, and then I'd have an icoseles triangle (AD=AC=r) where I know all the angles and two sides, and so can calculate the third side. Then, since I'd know the length of CD and I know the ratio of the lengths of the sides of BCD, I'd be done.

But BD doesn't pass through A. What I need to work out is if it's possible to move forward from here, to work out the length of BD with what is known.

If you want the story behind the problem, it's under this spoiler:
Spoiler:
I wanted to measure my hand span, but I didn't have a ruler or any kind of measuring device, so I decided to improvise. I had a standard DL size envelope, some pens, and some coins. I marked the ends on my fingers on the edge of the envelope, and used the coins to measure the remaining length of the envelope to work out my hand span. On one end, I had a coin whose diameter matched the length exactly, and so I could measure it easily. At the other end, I had no coins small enough to fit in the space, so I lined the diameter of the coin up with the edge of the envelope and traced the arc, figuring I could use it to derive the lengths I needed. This proved harder than I thought it would.
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gmalivuk
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Re: Finding the Length of a Chord

Postby gmalivuk » Sun Apr 10, 2016 1:04 pm UTC

How do you know from your setup that BC:BD::2:3?

Even with that fact, it seems underdetermined. For a given circle, there should be an entire curve of points for which the verticle line from the point to the circle is 2/3 as long as the horizontal line from the point to the circle.

If A=(0,0) and B=(x, y), all you know is (y - sqrt(r^2 - x^2))/(x - sqrt(r^2 - y^2)) = 2/3. I haven't bothered to plot it yet, but that looks to me like one equation with two unknowns.
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Carlington
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Re: Finding the Length of a Chord

Postby Carlington » Sun Apr 10, 2016 1:17 pm UTC

I didn't work it out geometrically. I improvised a compass which I used to find the intersection of BD and the circle with centre at B and radius BC. If that point is E, then BE appears to equal 2DE. Since it was close enough when I eyeballed it that it's not going to be a significant error on the scale of a handspan, I made BC:BD::2:3 an assumption in the problem, which is what I meant when I phrased it as "I have...". It might have been clearer to say "It is given that..."

Regarding the right angle of the triangle, that's the corner of the envelope I'm sketching on the back of, so as right as it gets. :P
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Re: Finding the Length of a Chord

Postby Flumble » Sun Apr 10, 2016 1:21 pm UTC

Does the same arc mirrored pass (nearly) through A? (preferably a 13th of the diameter/the distance between the center points short) If so, the arc subtends about 120°, or about 2 rad (114,59..°) at which the arc length is equal to the diameter.
You can of course also estimate how many of those arcs fit in a full revolution.

Considering the span of your hand can differ quite a bit and coins are pretty small, I'd say that the error in your estimation is considerably smaller than the error in your measurement.

B, as it stands, is just some point on a circle with CD as cross section. Like you said, only if one of the sides were to pass through A, would your construction give an answer.

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Re: Finding the Length of a Chord

Postby gmalivuk » Sun Apr 10, 2016 1:36 pm UTC

Carlington wrote:I didn't work it out geometrically. I improvised a compass which I used to find the intersection of BD and the circle with centre at B and radius BC. If that point is E, then BE appears to equal 2DE. Since it was close enough when I eyeballed it that it's not going to be a significant error on the scale of a handspan, I made BC:BD::2:3 an assumption in the problem, which is what I meant when I phrased it as "I have...". It might have been clearer to say "It is given that..."

Regarding the right angle of the triangle, that's the corner of the envelope I'm sketching on the back of, so as right as it gets. :P
Well, it's still insufficient to determine BD.

Edit:
circleplot.png
The curve through the middle of the circle is all the points B where BC:BD::2:3.
circleplot.png (17.34 KiB) Viewed 2523 times
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lorb
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Re: Finding the Length of a Chord

Postby lorb » Sun Apr 10, 2016 2:09 pm UTC

The problem is very clearly underdetermined.

You have a triangle BCD that has no length given anywhere. The fact that it's right, and that its hypotenuse happens to be the chord of some circle gets you nowhere, because every line can be the chord of a lot of circles and there is nothing that makes this circle any special.
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Re: Finding the Length of a Chord

Postby gmalivuk » Sun Apr 10, 2016 2:19 pm UTC

lorb wrote:The problem is very clearly underdetermined.

You have a triangle BCD that has no length given anywhere. The fact that it's right, and that its hypotenuse happens to be the chord of some circle gets you nowhere, because every line can be the chord of a lot of circles and there is nothing that makes this circle any special.
Yeah, the 2/3 ratio is the only thing that actually helps, because it narrows it down to a single curve instead of allowing B to be anywhere within the circle.
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Carlington
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Re: Finding the Length of a Chord

Postby Carlington » Sun Apr 10, 2016 2:24 pm UTC

No, I can see now very clearly that there's not enough here to do much of anything with. I had initially hoped that there was some deep geometric wizardry I just wasn't aware of or had forgotten that would result in something dependent on the radius or the diameter, or some sort of trick to do with the ratios of the sides of the triangle, but once gmal actually put it down as (y - sqrt(r^2 - x^2))/(x - sqrt(r^2 - y^2)) = 2/3 it clicked into place and I got what was being said.
Kewangji: Posdy zwei tosdy osdy oady. Bork bork bork, hoppity syphilis bork.

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Re: Finding the Length of a Chord

Postby gmalivuk » Sun Apr 10, 2016 2:41 pm UTC

Best bet is probably to just break a finger or two so you don't need to use the second coin at all.
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