Assignment 6
AX = XY=YB

by
Jenny Johnson

In this exploration, we are given three non-collinear points A, B and C.  We connect A to C and B to C to form the following figure.

The point of the exploration is to find a point X on AC and a point Y on BC such that AX = XY = YB.  So we want to find the points X and Y so the three segments mentioned are congruent.  An example of points X and Y are shown in the following diagram.

My first exploration to find the points X and Y was to draw circles. Any two circles with the same radius and centers at A and B will give us respective lengths of potential AX and BY such that AX = YB.  So, I drew a circle centered at A with an arbitrary radius.  The intersection of the circle with the segment AC I labeled X.  Then, I constructed a circle centered at B with radius AX.  The intersection of this circle with segment BC I labeled Y.  Thus, by construction AX = YB.

Now I can drag X along the line segment AC and the point Y also adjusts so that AX = YB.  I then constructed a circle centered at X with radius XA.

I can drag X until the circle centered at X with radius XA intersects the point Y.  Then the radius of the circle is XY, which means XY = XA = YB at this point.  A picture of this is shown below.

Thus, we have found a point X and a point Y such that AX = XY = YB.