Henry Ernest Dudeney/Puzzles and Curious Problems/228 - The Ladder/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $228$

The Ladder
A ladder was fastened on end against a high wall of a building.
It was unfastened and pulled out $4$ yards at the bottom.
It was then found that the ladder had descended just one-fifth of the length of the ladder.
What was the length of the ladder?


Solution

$6 \tfrac 2 3$ yards.


Proof

Let the length of the ladder be $L$ yards.

The ladder, the wall and the ground form a right triangle.

Hence by Pythagoras's Theorem:

\(\ds 4^2 + \paren {L - \dfrac L 5}^2\) \(=\) \(\ds L^2\)
\(\ds \leadsto \ \ \) \(\ds 4^2 + \dfrac {16 L^2} {25}\) \(=\) \(\ds L^2\)
\(\ds \leadsto \ \ \) \(\ds 400\) \(=\) \(\ds 9 L^2\) simplifying
\(\ds \leadsto \ \ \) \(\ds 20\) \(=\) \(\ds 3 L\) taking square root of both sides
\(\ds \leadsto \ \ \) \(\ds L\) \(=\) \(\ds 6 \tfrac 2 3\)

$\blacksquare$


Sources

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