ROTATING TRIANGLE PROBLEMS
1. What is the locus of the third vertex of a triangle when the
other two vertices are moved along the x-axis and y-axis, respectively.
You may want to cut out triangles and experiment by physically
moving the triangles along the x and y axes, marking the location
of the third vertex as you do so. Be sure to try it for different
shapped triangles.
generate an ellipse for non-zero a and b, a
not equal to b. Where are the foci?
In particular, if (a,0) is a point between 0 and 1 on the
x-axis, then the locus of that point as the segment of
unit length is moved with its ends on the x-axis and the y-axis
is an ellipse defined by
Note that a can be any real number, positive or negative
but degenerate ellipses are formed when a = 0 or a
= 1 (a segment along the respective axes) and a circle is formed
for a = .5.
generate an ellipse (or a degenerate ellipse) for all values
of a and h, where h is the altitude of a
triangle in initial position with a side of length 1 along the
x-axis and (a,0) is the point at the foot of the altitude.
The vacuum grinder is a toy for keeping executives busy and
therefore not interfere with the work of those they supervise.
Moving the handle, the attached points slide along the two slots
at right angles. Most of the time if you ask someone operating
the vacuum grinder "What path is traced out by the handle?"
they will say "It moves in a circle."
Being more aware, as mathematics teachers, we know the path is
an ellipse. Further, fixed points on the handle other than the
pivot points (these move along the axes) and the midpoint between
the axes (it moves in a circle) will have an ellipse for its locus.
A picture framing shop may have a device that works on the same
principle for cutting ovals out of matting stock. The cutting
knife can be adjusted to points on the handle in order to cut
the appropriate size oval.
I sometimes illustrate the locus of the third vertex on a triangle
by pasting a cut-out of a triangle on my vacuum grinder so that
two of the vertices match the pivot points. Then as the handle
is turned the locus of the third vertex can be watched.