Assignment 6 Write-up
6. Given three points A, B, and C. Draw a line intersecting AC in the point X and BC in the point Y such that
AX = XY = YB
The
most important thing to realize about this problem is that you
are going to have to use the idea of similar triangles. First
of all, let's start by drawing line segments AC and BC.
Now,
we need to put in a random point D on AC. (Notice that this is
not the X point that we will need for the answer, but it is a
point that we will use for similar triangles.) Let's also create
a circle with D as the center and with A lying on the circle.
Next,
we need to create a line passing through D that is parallel to
BC. Let's label the point that intersects the line and circle
as E. Also, let's go ahead and construct AB.
Notice
that AD and DE both have the same length since they are radii
of the same circle. Now, let's construct a circle with center
E and radius DE. Let's label the intersection of this circle and
AB as F, and let's construct segments, AD, DE, and EF.
Notice
that EF has the same length as the other two segments. Now let's
create a rhombus with DE and EF being two of the sides. Let's
also get rid of some of the "junk" that we don't need.
Can
you see that AD = DG = GF? If only we had been given line HF instead
of line BC! How can we use this picture to find the missing X
and Y points? The answer is . . . . similar triangles!!!
By
constructing parallel lines where applicable, we have constructed
two similar triangles, ADG and AXY. So since AD = DG, we know
that AX = XY. Also, notice that triangle AGF is similar to triangle
AYB. So GF is similar to YB. And since AD = DG = GF, we know that
AX = XY = YB!! Click here to see for yourself!!