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.
Dudeney-Modern-Puzzles-134.png
What is the distance in exact feet from the middle tree to the gate on the road?


Solution

$35$ feet.


Proof

Dudeney-Modern-Puzzles-134-solution.png

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

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