Samantha’s square courtyard
(total area less than 100 m^2
) has an area equal to an integer number
of square meters. She decides to
install an octagonal fish pond in the
courtyard.
To mark the sides of the
pond, she draws lines from each
corner of the square to the midpoints
of the two sides not touching said
corner. She finds that the perimeter
of the pond, thus delimited, is an
integer number of meters.
What is
the area of the courtyard, and what
is the perimeter of the pond?
Start out with a 10 by 10 square on an x-y plane, with the bottom left corner at (0,0).
To figure out the ratio of octagon to square sides, consider three lines.
The line extending from (0,10) to (5,0) has equation y = 10-2x.
The side of an octagon is formed by its intersections with
a) The line extending from (0,0) to (5,10)), which is y = 2x
b) The line extending from (0,5) to (10,0)), which is y = 5 - x/2
Using simple algebra, y= 2x and y=10-2x intersect at (5/2, 5).
Similarly, y = 5-x/2 and y=10-2x intersect at (10/3,10/3).
The distance between those two points is one side of the octagon, and it is (5/6)*sqrt(5) using pythagorean theorem.
The smallest possible octagon side (if the perimeter is an integer) is 1/8, so multiply all lengths by 1/8*(6/5)*(1/sqrt(5))
This makes the square side 10/8*(6/5)*(1/sqrt(5)) = (3/10)sqrt(5)
Then the area is 45/100, which is not an integer.
Scale all sides up by multiplying by 10, and the square area becomes area 45 and the the pool perimter 10.
Look ma, no trig!
