Given three segments of length 1, a and b in the plane, how can one construct segments of length a+b, |a-b|, ab, a/b, √a using ruler and compass? Which other calculator functions can be performed by geometric construction?

a+b

Extend the segment a to a reasonably long length. Measure segment b with the compass and lay off the same distance from the end of segment a along the line.

a-b

Measure segment b with the compass and lay off the same distance from the end of segment a back toward the other end of segment a.

ab

Construct a point, p, 1 unit away from segment a along some perpendicular to segment a, say the perpendicular bisector for sake of argument.  Lay off a segment of that perpendicular, starting at point p the same length as b.  Construct a perpendicular line at the other end of that segment from p, so it's parallel to a. Connect a ray from point p through each endpoint of segment a, extended at least to the newly constructed line. The points of intersection mark off a length of ab.

a/b

Construct a point, p, a distance b along a perpendicular, say perpendicular bisector for sake of argument. Construct a line parallel to a, at unit distance from p. Connect point p to each end  point of segment a by a new segment.  The two new segments' intersections with the parallel line mark a length of a/b.

Posted by Charlie on 2006-11-24 10:49:53