The triangular piece of land that was not for sale contains exactly
eleven acres. Of course it is not difficult to find the answer if we
follow the eccentric and tricky tracks of intricate trigonometry; or I
might say that the application of a well-known formula reduces the
problem to finding one-quarter of the square root of (4 × 370 × 116)
-(370 + 116 - 74)²--that is a quarter of the square root of 1936, which
is one-quarter of 44, or 11 acres. But all that the reader really
requires to know is the Pythagorean law on which many puzzles have been
built, that in any right-angled triangle the square of the hypotenuse is
equal to the sum of the squares of the other two sides. I shall dispense
with all "surds" and similar absurdities, notwithstanding the fact that
the sides of our triangle are clearly incommensurate, since we cannot
exactly extract the square roots of the three square areas.
[Illustration:
A
|\
| \.
| \ .
|5 \ .
| 7 \ .
E +--------- +C .
| | ` . .
| | `. .
|4 |4 ` . .
| 7 | ` ..
D----------+----------------- B
F
]
In the above diagram ABC represents our triangle. ADB is a right-angled
triangle, AD measuring 9 and BD measuring 17, because the square of 9
added to the square of 17 equals 370, the known area of the square on
AB. Also AEC is a right-angled triangle, and the square of 5 added to
the square of 7 equals 74, the square estate on A C. Similarly, CFB is a
right-angled triangle, for the square of 4 added to the square of 10
equals 116, the square estate on BC. Now, although the sides of our
triangular estate are incommensurate, we have in this diagram all the
exact figures that we need to discover the area with precision.
The area of our triangle ADB is clearly half of 9 × 17, or 76½ acres.
The area of AEC is half of 5 × 7, or 17½ acres; the area of CFB is half
of 4 × 10, or 20 acres; and the area of the oblong EDFC is obviously 4 ×
7, or 28 acres. Now, if we add together 17½, 20, and 28 = 65½, and
deduct this sum from the area of the large triangle ADB (which we have
found to be 76½ acres), what remains must clearly be the area of ABC.
That is to say, the area we want must be 76½ - 65½ = 11 acres exactly.