ABCDEF is a regular hexagon with sides of length h units. A square PQRS is drawn inside the hexagon, with PQ parallel to AB, SR parallel to ED, and vertices PQRS lying on FA, BC, CD and EF respectively.
The area of PQRS is 4 square units.
What is the exact value of h?
Let's get the ratio of the hexagon's side length to the square's side length. So let's make the hexagon's sides 1 unit.
Put vertices at (-1,0),(-1/2,sqrt(3)/2),(1/2,sqrt(3)/2),(1.0). The centers of both the hexagon and the square will be at the origin.
The equation of the line from (-1,0) to (-1/2,sqrt(3)/2) is
y = (x+1) * sqrt(3)
The corners of the square will be at y = +/- x.
(x+1) * sqrt(3) = - x (for the second quadrant so x is negative and y positive.
x * (1+sqrt(3)) = -sqrt(3)
x = -sqrt(3) / (1+sqrt(3))
Due to the vertical and horizontal symmetries about the origin, the full side length of the square is 2 * sqrt(3) / (1+sqrt(3)) relative to the side length of the hexagon.
We need the opposite, or reciprocal: The hexagon's side length is
(1+sqrt(3)) / ( 2 * sqrt(3)) times the square's side length, or about 0.788675134594813.
Since the square's side length is 2, h is twice this, or (1+sqrt(3)) / sqrt(3) =~ 1.57735026918963.
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Posted by Charlie
on 2024-07-22 12:38:45 |
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