Triangle With Fixed Base Moving Along the x-axis and the y-axis


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.

2. Show that the parametric equations

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.

3. Show that the parametric equations

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.

4. The vacuum grinder

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.

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