Heron of Alexandria/Problems/Rectangles
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Problem
- Find two rectangles with integral sides, such that:
Solution
The rectangles are:
- $53 \times 54$
and:
- $318 \times 3$
Proof
As can be seen:
\(\ds 53 \times 54\) | \(=\) | \(\ds 2862\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times \paren {318 \times 3}\) |
and:
\(\ds 3 \times \paren {2 \times \paren {53 + 54} }\) | \(=\) | \(\ds 642\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times \paren {318 + 3}\) |
and it is seen that the two rectangles fulfil the given conditions.
Let the given ratio be made general, that is, $n$ rather than $3$.
Let $u, v$ and $x, y$ be the sides of the rectangles.
Then we can write:
\(\ds u + v\) | \(=\) | \(\ds n \paren {x + y}\) | ||||||||||||
\(\ds x y\) | \(=\) | \(\ds n u v\) |
Thus the general solution is:
\(\ds x\) | \(=\) | \(\ds 2 n^3 - 1\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds 2 n^3\) | ||||||||||||
\(\ds u\) | \(=\) | \(\ds n \paren {4 n^3 - 2}\) | ||||||||||||
\(\ds v\) | \(=\) | \(\ds n\) |
Setting $n = 3$ gives the solution.
$\blacksquare$
Sources
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Light Reflected off a Mirror: $21$