Example — finding the path, the rectangle

Example — Finding the path: the rectangle

Let us take up again the problem of the rectangle with perimeter $24$ and area $32$.

Reasoning backwards: we need $x$ and $y$. We have two equations in two unknowns, so the problem is “well posed”. We can:

• solve by substitution (from the first: $y=12-x$; we substitute into the second);
• or observe that $x$ and $y$ are the solutions of a single quadratic equation, by Viète’s formulae: $t^2 - 12\,t + 32 = 0$ (see the next chapter).

We have two paths: both lead to the same answer, but the second is more elegant. It is good to learn to recognise when it is available.

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Topics: Working method
Concepts: Backward reasoning · Solving strategy
Skills: Solving systems
People: François Viète (Vieta)