An equilateral triangle lies with its three vertices at respective distances from the origin as 1, 2 and 3 units. What is the length of each side of the triangle?
Let (x_1,y_1), (x_2,y_2), and (x_3,y_3) be the
vertices of the equilateral triangle and s its
side length. WOLOG let (x_1,y_1) = (1,0). Then
x_2^2 + y_2^2 = 4 (1)
x_3^2 + y_3^2 = 9 (2)
s^2 = (x_2  1)^2 + y_2^2 (3)
s^2 = (x_3  1)^2 + y_3^2 (4)
s^2 = (x_3  x_2)^2 + (y_3  y_2)^2 (5)
Combining equations 1 and 3 gives
s^2 = 5  2*x_2 (6)
Combining equations 2 and 4 gives
s^2 = 10  2*x_3 (7)
Combining equations 6 and 7 gives
x_3 = x_2 + 5/2 (8)
Combining equations 1, 2, 5, 6, and 8 gives
(x_2 + 1)^2*(2*x_2  5) = 0 (9)
The value x_2 = 5/2 does not satisfy
equation (1). Therefore,
x_2 = 1
and
s = sqrt(7)

Posted by Bractals
on 20070905 10:43:23 