Question 1190673
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A diagram....<br>
{{{drawing(400,400,-1,4,-1,4
,line(0,0,1,0),line(1,0,1.707,.707),line(1.707,.707,1.707,1.707),line(1.707,1.707,1,2.414),line(1,2.414,0,2.414),line(0,2.414,-.707,1.707),line(-.707,1.707,-.707,.707),line(-.707,.707,0,0)
,locate(0,-.1,A),locate(-.9,.8,B),locate(-.9,1.8,C),locate(0,2.7,D),locate (1,2.7,E)
,line(0,0,0,2.414),line(0,0,1,2.414),line(-.707,1.707,0,1.707),line(-.707,.707,0,.707)
,locate(.1,1.8,P),locate(.1,.8,Q)
)}}}<br>
The information you have is the diameter of the octagon, which is AE in the diagram.  The objective (presumably) is to determine the length of the side of the octagon in terms of the length of the diagonal.<br>
We can solve the problem the other way around -- finding the length of the diagonal in terms of the length of a side of the octagon.<br>
For simplicity, we can let the side length of the octagon be 1.  So AB=BC=CD=DE=1; and PQ is also 1.<br>
Triangles AQB and CPD are 45-45-90 right triangles, so AQ=PD=sqrt(2)/2.<br>
Then in triangle ADE the legs are lengths {{{1}}} and {{{1+sqrt(2)}}}, and the hypotenuse is the given diameter.<br>
Using the Pythagorean Theorem on that triangle...<br>
{{{d^2=1^2+(1+sqrt(2))^2}}}<br>
Use decimal approximations and a calculator gives us d=2.613 to 3 decimal places.<br>
So, given the diameter d, the side length of the octagon is d/2.613 = 0.383d.<br>