Henry Ernest Dudeney/Puzzles and Curious Problems/157 - Counting the Matches/Solution

by : $157$

 * Counting the Matches

Solution
First we recall Heron's Formula for the area of a triangle with sides equal to $a$, $b$ and $c$:


 * $\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$

where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.

With $36$ matches you can make a triangle with sides $17$, $10$, $9$ whose area $\AA$ is calculated by Heron's Formula to be:

With $30$ matches you can make a triangle with sides $13$, $12$, $5$, which is the $5$-$12$-$13$ Pythagorean Triangle whose area $\AA$ is calculated by Area of Triangle in Terms of Side and Altitude to be:

With $24$ matches you can make a triangle with sides $10$, $8$, $6$, which is the $3$-$4$-$5$ Pythagorean Triangle scaled up by a factor of $2$, whose area $\AA$ is calculated by Area of Triangle in Terms of Side and Altitude to be: