The three vertices of an equilateral triangle are at distances 1, 2, and 3 from a line.
Find all possible areas of this triangle.
Using Geometer's Sketchpad, when the distance-2 vertex is the sole point on its side of the line the area is about 10.9, but if the distance-3 is the isolated vertex the area is about 12.0. If the distance-1 vertex is the one isolated, the area is about 6.93 or 7,5 depending on the construction.
(The 6.93 is spurious; see below).
If all are on the same side, I get 1.73 as the area.
I suspect the other cases have alternatives also, but it's hard to see the appropriate construction. (again, see below -- the Matlab calculation)
Matlab calculation:
syms x1 x2 x3
for y1=[1 -1]
for y2=[2 -2]
for y3=[3 -3]
x = vpasolve([(x1-x2)^2+(y1-y2)^2==(x2-x3)^2+(y2-y3)^2, ...
(x1-x2)^2+(y1-y2)^2==(x1-x3)^2+(y1-y3)^2, ...
abs(atand((y3-y1)/(x3-x1))-atand((y2-y1)/(x2-x1)))==60], ...
[x1,x2,x3]);
if ~isempty(x.x1)
fprintf('(%9.6f,%2d) (%9.6f,%2d) (%9.6f,%2d) %9.6f %9.6f
', ...
x.x1,y1,x.x2,y2,x.x3,y3, ...
(sqrt(3)/4)*((x.x1-x.x2)^2+(y1-y2)^2), ...
(sqrt(3)/4)*((x.x1-x.x3)^2+(y1-y3)^2))
end
end
end
end
finds
vertices area based on
side 1 side 2
(-40.716919, 1) (-45.913071, 2) (-44.181020,-3) 12.124356 12.124356
(-25.356636, 1) (-21.315184,-2) (-20.737834, 3) 10.969655 10.969655
(-25.356636,-1) (-21.315184, 2) (-20.737834,-3) 10.969655 10.969655
(-40.716919,-1) (-45.913071,-2) (-44.181020, 3) 12.124356 12.124356
which seem to be the 10.9 and 12.1 found via GSP. The limitation would be arising from looking for a difference of 60.
Changing the sought difference in angles to 120° gives
(-43.026320, 1) (-45.913071,-2) (-41.871619,-3) 7.505553 7.505553
(-43.026320,-1) (-45.913071, 2) (-41.871619, 3) 7.505553 7.505553
which is the 7.5 found by GSP.
specifying "digits 10" before doing the vpasolve gives, when looking for 60°,
(-38.802870, 1) (-37.070819, 2) (-38.802870, 3) 1.732051 1.732051
(-40.716919, 1) (-45.913071, 2) (-44.181020,-3) 12.124356 12.124356
(-25.356636, 1) (-21.315184,-2) (-20.737834, 3) 10.969655 10.969655
(-25.356636,-1) (-21.315184, 2) (-20.737834,-3) 10.969655 10.969655
(-40.716919,-1) (-45.913071,-2) (-44.181020, 3) 12.124356 12.124356
(-38.802870,-1) (-37.070819,-2) (-38.802870,-3) 1.732051 1.732051
now finding the 1.7 solution from GSP.
Using angle difference of 120° with digits 10:
(-43.026320, 1) (-44.758370, 2) (-43.026320, 3) 1.732051 1.732051
(-43.026320, 1) (-45.913071,-2) (-41.871619,-3) 7.505554 7.505553
(-43.026320,-1) (-45.913071, 2) (-41.871619, 3) 7.505554 7.505553
(-43.026320,-1) (-44.758370,-2) (-43.026320,-3) 1.732051 1.732051
The Matlab program has found
1.732051
7.505553
10.969655
12.124356
What about GSP's 6.93?
Going back, I see that I mishandled the case where the area was 6.93. I confused two lines and the distances of the vertices from the line were 1, 1, and 3 rather than 1, 2 and 3.
Edited on March 29, 2024, 1:31 pm
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Posted by Charlie
on 2024-03-29 13:30:11 |
computer solution
