Question 1123406
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Picture the octagon as the 26-inch square with the corners cut off at 45-degree angles.  The cut-off corners are 45-45-90 right triangles; the hypotenuse of each of those triangles is a side of the octagon.<br>
Then the 26-inch distance across the square consists of one side of the octagon, plus the legs of two of the triangles.<br>
In a 45-45-90 right triangle, each leg is (1/sqrt(2)) times the length of the hypotenuse.  So if x is the length of a side of the octagon, the 26-inch side of the octagon is<br>
{{{((1/sqrt(2))*x) + (x) + ((1/sqrt(2))*x)}}}
or
{{{x(1+sqrt(2))}}}<br>
Then the side length of the octagon is<br>
{{{26/((1+sqrt(2)))}}}<br>
which to the nearest hundredth of an inch is 10.77 inches.