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?
I drew a large square to scale and then drew in the 8 line segments. The vertices of the the inner octagon were identified. Assuming a unit square and later scaling everything to side length 'S', I found coordinates of each point using the equations of each of 8 lines, at least until a pattern emerged.
The vertices:
A(1/2,1/4)
B(2/3,1/3)
C(3/4,1/2)
D(2/3,2/3)
E(1/2,3/4)
F(1/3,2/1)
G(1/4,1/2)
H(1/3,1/3)
It turns out every line segment has length √5/12, so the perimeter is 8*√5*S/12 = 2*√5*S/3 = P
For this perimeter to be an integer, S must be a multiple of 3√5, call it 3k√5 .
P = 2*√5*(3k√5)/3 = 10k
Area = (3k√5)^2 = 45k^2. But area is <100 so k=1
Area = 45
Perimeter of octagon = 10
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Posted by Larry
on 2024-12-23 22:10:33 |
Analytic solution
