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#1




Math Quiz 11
There are nine tiny dots, labeled 1 thru 9, on a gym floor. Jane measures the distances between pairs as follows: Code:
2 4 47.8017 3 5 69.2026 5 9 148.492 7 8 86.764 5 6 63.0714 2 8 189.528 6 7 65.192 4 9 147.085 2 3 90.5207 6 8 86.0349 3 9 171.956 4 7 88.8144 
#2




Re: Math Quiz 11
I plugged all the points and dimensions into Solidworks, nothing is defined so my answer is:
Unsolvable! (99% sure I did something wrong) 
#3




Re: Math Quiz 11
8.06 units?
Also, no information is given about point number 1. Is that a typo? 
#4




Re: Math Quiz 11
157.674

#5




Re: Math Quiz 11
294.325?

#7




Re: Math Quiz 11
I think you have two different points for #2.

#8




Re: Math Quiz 11
Quote:
EDIT  Ignore me. Last edited by Cothron Theiss : 05052017 at 09:46 AM. 
#9




Re: Math Quiz 11
That's the distance to the top right point, not the one the number is closest to.

#10




Re: Math Quiz 11
Woops! Thanks for catching that. You're right.

#11




Re: Math Quiz 11
174.7256.
Link to my CAD (I found this cool, CAD package online in order to do it... seems to work pretty well, at least for this stuff, and you can have a public free account!) https://cad.onshape.com/documents/43...9620e77d130f51 Anyone see any mistakes? I even labeled my points Edit: Out of curiosity Ether, is there a purely mathematical way to arrive at the answer, one that can be done by hand on a single sheet of paper? Last edited by Jon Stratis : 05052017 at 09:52 AM. 
#12




Re: Math Quiz 11
Looking at the topology, we have two triangles: 359 and 867.
Points 6 and 5 are connected to each other, points 3 and 8 are connected to point 2, 7 and 9 are connected to point 4, and points 2 and 4 are connected. That is: Code:
9  4  7  \  /   3  2  8   / \  5  6 
#13




Re: Math Quiz 11
Quote:
I figured you were lurking in the background, waiting to pounce on this. Here's another way to analyze this: Jane made 12 distance measurements. Pick a Cartesian coordinate system with point2 at the origin and point3 on the +X axis. So the coordinates of point2 are [0,0]. Since Jane measured the distance from 2>3, and point3 lies on the +X axis, the coordinates of point3 are [146,0]. That leaves 6 points (4 thru 9) whose coordinates are unknown. Since there are 2 scalar values per point, you have 12 unknowns. But you only have 11 measurements left, because you already used the distance from 2>3. So you have an underdetermined system of nonlinear equations. There is no unique solution. Quote:
Quote:
Last edited by Ether : 05052017 at 11:13 AM. 
#14




Re: Math Quiz 11
I knew I should have posted my musings last night that this wasn't fully constrained! I had gotten nowhere trying to figure it out by hand, and resolved to give CAD a try this morning. When it came back with an answer, I figured why not

#15




Re: Math Quiz 11
Quote:
Edit  or possibly as many as four. I forgot to consider folding the polygon 5876 over segment 86. Edit2  to clarify, by "edge cases" I mean singularity in the system of equations. Edit3  Perhaps even more, due to which way the angles at points 2 and 4 bend. Let's go with this: With the length of 58 determined, and excluding singularity cases, there would be a (fairly small) finite number of solutions. This arises from the nonlinear nature of the system of equations. For example, the singleunknown equation x^{5}  5x^{3} + 4x = 0 has exactly five real solutions. This set of equations appears to be second order (to simplify, square all of the measured lengths to eliminate those square roots), but the cross terms among the different unknowns increases the number of possible solutions. Edit4  I think I have a way to find the solutions when the length of segment 58 is given, for all 16 possibilities of the four remaining ambiguities* in a nottoocomplex excel spreadsheet, if you don't mind looking for crossing points on graphs. This is stretching Jon's question, but not totally breaking it. I'm going to work this up with a 58 distance of 55.17 (to commemorate the date of 5 May 2017), though this will be adjustable if this turns out not to be an interesting value. If I have enough energy left, I will generate all the plots in excel as well**. Ether, if you have a different 58 distance in mind for 58, please let me know. * At a geometric level, these ambiguities arise from the law of cosines: c^{2} = a^{2} + b^{2} + 2abcosγ. This equation is ambiguous in that while a and b are always positive, γ may be either positive or negative, and cosγ = cos(γ). ** As it turns out, the graph of the initial set of distances Ether gave is a "doubledown" on the classic Bridges of Konigsberg problem  every one of the eight vertices has three measurements! This will make drawing lines using excel scatter plot more interesting. Last edited by GeeTwo : 05052017 at 07:31 PM. 
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