Constructing the Cube Root of Two
Constructing the Cube Root of Two
This Demonstration shows a construction of using a ruler with two marks (red and black ) at a distance 1 unit apart, constrained to lie on the axis and the blue line. Drag point until the straight line touches the hexagon at . Then .
3
2
A
B
x
B
AB
C
|BC|=
3
2
Explanation: The slope of is and the slope of the blue line is . Since their product is , the blue line is perpendicular to the line . Therefore is a right-angled triangle, so =1+. The angle is , so . Since triangles and are similar, . Therefore =3, substitute for to get =1+3, and clear the denominator: (1+y)=+3. Expand to get +2+=1+2y++3, rearrange terms to get +2-2y-4=0, and factor (y+2)-2(y+2)=0, ; and take the positive solution .
EC
3
-
3
3-1
EC
BEC
2
y
2
z
DEB
π/6
|DB|=zsinπ/6=z/2
AFC
ADB
1+y=
3
2(z/2)=3
z2
z
2
(1+y)
2
z
2
y
2
(1+y)
2
2
y
2
(1+y)
2
y
3
y
4
y
2
y
4
y
3
y
3
y
-2(y+2)=0
3
y
y=
3
2