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Cartesian Pattern

According to Pólya, the Cartesian pattern is the resolution method for arithmetical or geometrical problems based on equations. The first step is to translate the question into one or more algebraic equalities, which express the relationship between the numerical data (the coefficients) and the quantities to be determined (the unknowns). This relationship can be described in text, or be depicted in a figure.

The second step is to solve the equations.

Normally, the quantity requested by the problem is only one, which permits us to reduce the procedure to a single equation, whose sides contain two different expressions of the same quantity. Consider, for example, a problem asking for one of the legs of a right triangle given that the length of this leg is half the length of the hypotenuse and that the other leg has length 1. If the unknown leg is denoted by and the hypotenuse by , then

 (1)

as specified, and, moreover,

 (2)

by the Pythagorean theorem. These equalities give rise to the equation

 (3)

where both sides are equal to the length of the hypotenuse. Solving with respect to gives

 (4)

This is the length of the unknown leg. Substituting this number in the first (or in the second) equation we can also derive the length of the hypotenuse, which is

Cartesian, Pattern of Two Loci

This entry contributed by Margherita Barile

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References

Pólya, G. Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, 2 vols. in One. New York: Wiley, 1981.

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Cartesian Pattern

Cite this as:

Barile, Margherita. "Cartesian Pattern." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CartesianPattern.html