Henry Ernest Dudeney/Puzzles and Curious Problems/242 - Correcting a Blunder/Solution 2
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Puzzles and Curious Problems by Henry Ernest Dudeney: $242$
- Correcting a Blunder
- Mathematics is an exact science, but first-class mathematicians are apt, like the rest of humanity, to err badly on occasions.
- On referring to Peter Barlow's Elementary Investigation of the Theory of Numbers, we hit on this problem:
- "To find a triangle such that its three sides, perpendicular, and the line drawn from one of the angles bisecting the base
- may all be expressed in rational numbers."
- "To find a triangle such that its three sides, perpendicular, and the line drawn from one of the angles bisecting the base
- He gives as his answer the triangle $480$, $299$, $209$, which is wrong and entirely unintelligible.
- Readers may like to find a correct solution when we say that all the five measurements may be in whole numbers,
- and every one of them less than a hundred.
- It is apparently intended that the triangle must not itself be right-angled.
Solution
Proof
The triangle in solution is the obtuse triangle whose sides are $66$, $41$ and $85$.
We have:
- $40^2 + \paren {66 + 9}^2 = 85^2$, demonstrating that the extended triangle is right-angled
- $9^2 + 40^2 = 41^2$, demonstrating that the triangle on the far right is right-angled
- $40^2 + \paren {\dfrac {66} 2 + 9}^2 = 58^2$, demonstrating that the triangle formed from the two on the right is right-angled
$\blacksquare$
Sources
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $280$. Correcting a Blunder