Ken's Puzzle of the Week
Square Root of a Line
- Given a compass, a straightedge, and a line segment AB of length 3, draw a
line segment with a length of the square root of 3.
- Given a compass, a straightedge, a line segment AB of length "x", and a
unit line segment YZ, draw a line segment with a length sqrt(x).
Source: Thane Larson, quoting Descartes.
Solutions were received from Mark Rickert, David Madfes, Philippe
Fondanaiche, Dan Chirica, K Sengupta, Frak Mullin, Dlaudio Baiocchi, Joseph
DeVincentis, Alan O'Donnell, Kirk Bresniker, Marcus Macrae, Bernie R.
Erickson, Bernd Jedrissek.1. While part 1 can be solved using part 2,
many solvers found faster methods.
From many (thanks to Philippe Fondanaiche for the picture):
From Claudio Baiocchi:
2. (From Mark Rickert):
Given a
point, P, in a circle and a line through the point, let A and B be the
points where the line intersects the circle. The products of the lengths
of the line segments AP and BP is a constant.
[So to
solve this puzzle:] Create a
line segment of length x+1 (AB) with point C dividing AB into sections
of length x (AC) and 1 (BC).
Bisect it
(center is P), and draw a circle with AB as the diameter and P as the
center.
Create a
line through point C that is perpendicular to AB. It will intersect the
circle at points D and E. Note that CD=CE.
From above
CD*CE = AC*BC = x*1.
Therefore,
CD=CE=sqrt(x).
Some sites for square-root of a line:
http://www.egge.net/~savory/maths9.htm
http://www.cs.cas.cz/portal/AlgoMath/Geometry/PlaneGeometry/GeometricConstructions/SquareSquareRootConstruction.htm
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