In a circle of radius r there are two parallel chords.
The distance between the chords is equal to the average of the lengths of the chords.
Find the relation between the distances from the chords to the center of the circle.
syms x1 y1 x2 y2
clc
for y1=.98:-.02:0
x1=sqrt(1-y1^2);
eq1=x2^2+y2^2==1;
eq2=y1-y2==x1+x2;
eqs=[eq1, eq2];
s=solve(eqs,y2,x2);
x2n=eval(s.x2); % solutions are 1x2 matrices for each variable
y2n=eval(s.y2); % as both left and right ends are found
[num ix]=max(x2n); % max finds the positive x2 value
x2n=x2n(ix);
y2n=y2n(ix);
fprintf('%4.2f %7.5f %7.5f %4.2f\n',y1,x1,y2n, x2n);
end
produces a table of values for such pairs of chords for a unit circle centered on the origin. Multiply by r for the general case, as generality is not lost. The chords were chosen to be horizontal.
The first column gives the y coordinate of the top chord of the two. The next is the x coordinate of the right-hand end of the top chord, and is thus half the length of that chord. The last two columns are the same data for the bottom chord.
The table shows that the distance of each chord from the center is half the length of the other chord.
Top chord Bottom chord
right-end right-end
y x y x
0.98 0.19900 -0.19900 0.98
0.96 0.28000 -0.28000 0.96
0.94 0.34117 -0.34117 0.94
0.92 0.39192 -0.39192 0.92
0.90 0.43589 -0.43589 0.90
0.88 0.47497 -0.47497 0.88
0.86 0.51029 -0.51029 0.86
0.84 0.54259 -0.54259 0.84
0.82 0.57236 -0.57236 0.82
0.80 0.60000 -0.60000 0.80
0.78 0.62578 -0.62578 0.78
0.76 0.64992 -0.64992 0.76
0.74 0.67261 -0.67261 0.74
0.72 0.69397 -0.69397 0.72
0.70 0.71414 -0.71414 0.70
0.68 0.73321 -0.73321 0.68
0.66 0.75127 -0.75127 0.66
0.64 0.76837 -0.76837 0.64
0.62 0.78460 -0.78460 0.62
0.60 0.80000 -0.80000 0.60
0.58 0.81462 -0.81462 0.58
0.56 0.82849 -0.82849 0.56
0.54 0.84167 -0.84167 0.54
0.52 0.85417 -0.85417 0.52
0.50 0.86603 -0.86603 0.50
0.48 0.87727 -0.87727 0.48
0.46 0.88792 -0.88792 0.46
0.44 0.89800 -0.89800 0.44
0.42 0.90752 -0.90752 0.42
0.40 0.91652 -0.91652 0.40
0.38 0.92499 -0.92499 0.38
0.36 0.93295 -0.93295 0.36
0.34 0.94043 -0.94043 0.34
0.32 0.94742 -0.94742 0.32
0.30 0.95394 -0.95394 0.30
0.28 0.96000 -0.96000 0.28
0.26 0.96561 -0.96561 0.26
0.24 0.97077 -0.97077 0.24
0.22 0.97550 -0.97550 0.22
0.20 0.97980 -0.97980 0.20
0.18 0.98367 -0.98367 0.18
0.16 0.98712 -0.98712 0.16
0.14 0.99015 -0.99015 0.14
0.12 0.99277 -0.99277 0.12
0.10 0.99499 -0.99499 0.10
0.08 0.99679 -0.99679 0.08
0.06 0.99820 -0.99820 0.06
0.04 0.99920 -0.99920 0.04
0.02 0.99980 -0.99980 0.02
0.00 1.00000 -1.00000 0.00
Note the degenerate limiting case at the bottom, where the top chord is a diameter of the circle and the bottom chord has been reduced to a point so that the distances are zero and 1 respectively (units being the radius).
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Posted by Charlie
on 2022-02-10 09:38:22 |
computer-aided solution
