Henry Ernest Dudeney/Puzzles and Curious Problems/242 - Correcting a Blunder/Solution 2

Puzzles and Curious Problems by Henry Ernest Dudeney: $242$

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."

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.

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

The side whose length is $58$ is seen to bisect the base.

The area of the triangle in solution is $\dfrac 1 2 \times 40 \times 66 = 1320$.

$\blacksquare$