Problem:
It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the and axes and so that the medians to the midpoints of the legs lie on the lines and . The number of different constants for which such a triangle exists is
Answer Choices:
A.
B.
C.
D.
E. more than
Solution:
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In any right triangle with legs parallel to the axes, one median to the midpoint of a leg has slope times that of the other. This is easily shown by analytic geometry: any triangle of this sort may be labelled as in the figure, where are arbitrary except . (The figure has the right angle in the lower right, but the labelling allows complete generality - why?) Note that the slopes of the medians are
Therefore, in our problem is either 12 or .
In fact, both values are possible, each for infinitely many triangles. We show this for . Take any right triangle having legs parallel to the axes and a hypotenuse with slope , e.g., the triangle with vertices . Then the medians to the legs have slopes and (Why?). Now translate the triangle (don't rotate!) so that its medians intersect at the point where and intersect. This forces the medians to lie on these lines (Why?). Finally, for any central dilation of this triangle (a larger or smaller triangle with the same centroid and sides parallel to this one's sides), the medians will still lie on these lines.