Projects - Cengage Learning

C-88 Appendix C
PROJECT
Geometry Workbooks on the Euler Line and the Nine-Point Circle
The exercises in this project form a unit introducing two remarkable geometric
results. Workbook I uses a specific example to introduce the Euler line and the
nine-point circle of a triangle. Workbook II develops the Euler line for the general
case. The exercises are straightforward in that they involve only basic coordinate
geometry and systems of two linear equations. However, a good deal
of algebraic calculation is required in both workbooks, and just one error along
the way will spoil the result at the end of that workbook. Thus this is a good occasion
for group work, assuming that each person in a group takes an active
role in checking results. For both workbooks, one group can be assigned to
each exercise. Some exercises, however, require results from one or more previous
exercises. For instance, in Workbook I, Exercises 4–7 depend upon the
results of the previous exercises. So there is a need for some coordination here;
the class as a whole needs to be certain that the results in Exercises 1–3 are correct
before the groups assigned to Exercises 4–7 can begin their work. A similar
situation occurs in several places in Workbook 2.
Workbook 1
For this workbook you are given ^ABC, with vertices as indicated in the following
figure.
y
C(0, 6)
A(_4, 0) B(2, 0)
x
1. (a) A median of a triangle is a line segment drawn from a vertex to the midpoint
of the opposite side. For ^ABC, find the equation of the line containing
the median to side BC. Next, find the equation of the line
containing the median to side AC. Now, find the point where these two
medians intersect.
(b) A theorem from geometry states that in any triangle the three medians
are concurrent (i.e., intersect in a single point). Use this to check your
last answer in part (a) as follows. Find the equation of the line containing
the median to AB. Then check that this median passes through the
intersection point found in part (a).
Remark: For any triangle the point where the three medians intersect is
called the centroid of the triangle. It can be shown that the centroid is the
center of gravity or balance point of the triangle.
(c) Use a graphing utility to draw ^ABC along with the three lines
containing the medians. Check to see that the location of the centroid
appears to be consistent with coordinates obtained in part (a).