FARMER WURZEL'S ESTATE.
(
Patchwork Puzzles)
I will now present another land problem. The demonstration of the answer
that I shall give will, I think, be found both interesting and easy of
comprehension.
Farmer Wurzel owned the three square fields shown in the annexed plan,
containing respectively 18, 20, and 26 acres. In order to get a
ring-fence round his property he bought the four intervening triangular
fields. The puzzle is to discover what was then the whole area of his
estate.
Answer:
The area of the complete estate is exactly one hundred acres. To find
this answer I use the following little formula,
__________________
/4ab - (a + b + c) squared
--------------------
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 21/2, F is 41/2, 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 x 12 = 144 acres, and the portions 1, 2, 3, 4, not
included in the estate, have the respective areas of 121/2, 171/2, 91/2, and
41/2. These added together make 44, which, deducted from 144, leaves 100
as the required area of the complete estate.
Informational
