Problem:
How many non-congruent right triangles are there such that the perimeter in cm and area in are numerically equal?
Answer Choices:
A. none
B.
C.
D.
E. infinitely many
Solution:
We will show that the ratio of perimeter to area in a triangle can be changed at will by replacing the triangle by a similar one. Thus in each class of similar right triangles, there is one with perimeter (in whatever length units, say cm ) equal to area (in whatever area units), that is, with the ratio being . Since there are infinitely many non-similar right triangles, thus there are infinitely many non-similar (hence non-congruent) right triangles with perimeter area.
Let be the lengths (in cm ) of the two sides and the hypotenuse respectively of an arbitrary right triangle. The ratio of perimeter () to the area ( ) is
Now consider the similar triangle with sides . The ratio is now
Thus, choose and in the new triangle perimeter area.
Altemate solution sketch. If and are the lengths of the legs of a right triangle, then the area and perimeter are numerically equal if and only if
This is one equation in two (nonnegative) real variables. Generally such an equation has infinitely many solutions. Also, one expects different solutions to result in non-congruent triangles. Therefore, one should already believe that is correct. A complete analysis (subtle, but left to the reader!) shows that this belief is correct, even though solving the above equation (by rearranging and squaring to remove the radical) introduces extraneous roots, and even though some distinct pairs of valid solutions represent congruent triangles, e.g., and .