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 x 17, or 761/2 acres.  The area of AEC is half of 5 x 7, or 171/2 acres; the area of CFB is half of 4 x 10, or 20 acres; and the area of the oblong EDFC is obviously 4 x 7, or 28 acres.  Now, if we add together 171/2, 20, and 28 = 651/2, and deduct this sum from the area of the large triangle ADB (which we have found to be 761/2 acres), what remains must clearly be the area of ABC.  That is to say, the area we want must be 761/2 — 651/2 = 11 acres exactly.

190.—­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,

__________________
\/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.

[Illustration]

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.

191.—­THE CRESCENT PUZZLE.

Referring to the original diagram, let AC be x, let CD be x — 9, and let EC be x — 5.  Then x — 5 is a mean proportional between x — 9 and x, from which we find that x equals 25.  Therefore the diameters are 50 in. and 41 in. respectively.