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.

To explore this GSP file, click here.