Henry Ernest Dudeney/Modern Puzzles/134 - A Fence Problem/Solution
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Modern Puzzles by Henry Ernest Dudeney: $134$
- A Fence Problem
- A man has a square field, $60 \ \mathrm {ft.}$ by $60 \ \mathrm {ft.}$, with other property, adjoining the highway.
- For some reason he put up a straight fence in the line of three trees, as shown,
- and the length of fence from the middle tree to the tree on the road was just $91$ feet.
- What is the distance in exact feet from the middle tree to the gate on the road?
Solution
$35$ feet.
Proof
Let $x$ be the distance required.
Let $y$ be the distance marked on the diagram.
We have:
\(\text {(1)}: \quad\) | \(\ds x^2 + y^2\) | \(=\) | \(\ds 91^2\) | Pythagoras's Theorem | ||||||||||
\(\text {(2)}: \quad\) | \(\ds \dfrac x y\) | \(=\) | \(\ds \dfrac {60} {60 + y}\) | Definition of Similar Triangles | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \dfrac {60 x} {60 - x}\) | rearranging $2$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^2 + \paren {\dfrac {60 x} {60 - x} }^2\) | \(=\) | \(\ds 91^2\) | substituting for $y$ in $(1)$ |
The above leads to a quartic equation which is irksome to solve.
However, we are told that $x$ is an exact number of feet.
We have that $91^2$ is the sum of two square numbers in exactly one way:
\(\ds 91^2\) | \(=\) | \(\ds 8281\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7056 + 1225\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7056 + 1225\) | that is: $84^2 + 35^2$ |
Inserting $x = 35$ and $y = 84$ into $2$ proves that these numbers are consistent with the premises.
Hence the result.
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $134$. -- A Fence Problem
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $264$. A Fence Problem