Let ABCDEF be a regular hexagon with side 1. Point X, Y are on sides CD and DE respectively, such that the perimeter of DXY is 2. Determine angle XAY.
Angle XAY is 30°.
The program uses the law of cosines three times each for a variety of DX values: first to find DY from DX, then AX and AY to find XAD and YAD. These latter two are added together to find XAY.
It's based on a diagonal with a full 120° angle at the end centered on the diagonal's end, and then the angles above and below, each 60° relative to the diagonal. The two angles at the other end of the diagonal are added together to get the full angle XAY.
syms y
for x=.05:.05:.9
s=solve( x+y+sqrt(x^2+y^2-2*x*y*cosd(120))==2);
d1=sqrt(4+x^2-2*2*x*cosd(60));
A1=acosd((d1^2+4-x^2)/(2*d1*2));
d2=sqrt(4+s^2-2*2*s*cosd(60));
A2=acosd((d2^2+4-s^2)/(2*d2*2));
fprintf('%5.2f %17.15f %8.6f\n',[x eval(s) eval(A1+A2)])
end
The table shows that the answer is the same 30° regardless of how the 2 unit perimeter is achieved.
DX DY XAY
0.05 0.962025316455696 30.000000
0.10 0.923076923076923 30.000000
0.15 0.883116883116883 30.000000
0.20 0.842105263157895 30.000000
0.25 0.800000000000000 30.000000
0.30 0.756756756756757 30.000000
0.35 0.712328767123288 30.000000
0.40 0.666666666666667 30.000000
0.45 0.619718309859155 30.000000
0.50 0.571428571428571 30.000000
0.55 0.521739130434783 30.000000
0.60 0.470588235294118 30.000000
0.65 0.417910447761194 30.000000
0.70 0.363636363636364 30.000000
0.75 0.307692307692308 30.000000
0.80 0.250000000000000 30.000000
0.85 0.190476190476190 30.000000
0.90 0.129032258064516 30.000000
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Posted by Charlie
on 2024-05-26 13:09:10 |
solution
