(related to Problem: Farmer Wurzel's Estate)

The area of the complete estate is exactly one hundred acres. To find this answer I use the following little formula,

$$\frac{\sqrt{4ab - (a + b + c)^2}}{4},$$

where $a, b, c$ represent the three square areas, in any order. The expression gives the area of the triangle $A.$ This will be found to be $9$ acres. It can be easily proved that $A, B, C,$ and $D$ are all equal in area; so the answer is $26 + 20 + 18 + 9 + 9 + 9 + 9 = 100$ acres.


Here is the proof. If every little dotted square in the diagram represents an acre, this must be a correct plan of the estate, for the squares of $5$ and $1$ together equal $26;$ the squares of $4$ and $2$ equal $20;$ and the squares of $3$ and $3$ added together equal $18.$ Now we see at once that the area of the triangle $E$ is $2\frac 12,$ $F$ is $4\frac 12,$ and $G$ is $4.$ These added together make $11$ acres, which we deduct from the area of the rectangle, $20$ acres, and we find that the field $A$ contains exactly $9$ acres. If you want to prove that $B, C,$ and $D$ are equal in size to $A,$ divide them in two by a line from the middle of the longest side to the opposite angle, and you will find that the two pieces in every case, if cut out, will exactly fit together and form $A.$

Or we can get our proof in a still easier way. The complete area of the squared diagram is $12 \times 12 = 144$ acres, and the portions $1, 2, 3, 4,$ not included in the estate, have the respective areas of $12\frac 12,$ $17\frac 12,$ $9\frac 12,$ and $4\frac 12.$ These added together make $44,$ which, deducted from $144,$ leaves $100$ as the required area of the complete estate.

Thank you to the contributors under CC BY-SA 4.0!



Project Gutenberg

  1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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