Problem: A triangle has a fixed base that is inches long. The median from to side is inches long and can have any position emanating from . The locus of the vertex of the triangle is:
Answer Choices:
A. a straight line in. from .
B. a circle with as center and radius in.
C. a circle with as center and radius in.
D. a circle with radius in. and center in. from along .
E. an ellipse with as a focus.
Solution:
Perhaps the easiest approach to this problem is through coordinate geometry.
Let the triangle be and . Then , the midpoint of , has the coordinates . Therefore, using the distance formula, we have
or
This equation represents a circle with radius inches and center inches from along ;
or
as extreme values we may have and , using the notation with its usual meaning. From the median formula, with and , we have . Using the given extreme values in succession, we have and , and and . Therefore, the maximum distance between the extreme positions of is 6 inches. These facts, properly collated, lead to the same result as that shown above.