Answer:
7.46
Explanation:
To find the side of the octagon, we will use the triangles on the corner.
The legs of these triangles are (18 - x)/2 and the hypotenuse is x. So, using the Pythagorean theorem, we can write the following equation
[tex]x^2=(\frac{18-x}{2})^2+(\frac{18-x}{2})^2[/tex]
Then, we need to rewrite this expression
[tex]\begin{gathered} x^2=2(\frac{18-x}{2})^2 \\ \\ x^2=\frac{2(18-x)^2}{2^2} \\ \\ x^2=\frac{2(18-x)^2}{4} \\ \\ x^2=\frac{(18-x)^2}{2} \\ \\ 2x^2=(18-x)^2 \\ 2x^2=18^2-2(18)(x)+x^2 \\ 2x^2=324-36x+x^2 \\ 2x^2-324+36x-x^2=0 \\ x^2+36x-324=0 \end{gathered}[/tex]
Using the quadratic equation, we get that the solution is
[tex]\begin{gathered} x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ \text{ Where a = 1, b = 36, and c = -324. Then} \\ \\ x=\frac{-36\pm\sqrt{36^2-4(1)(-324)}}{2(1)} \\ \\ x=\frac{-36\pm\sqrt{2592}}{2} \\ \\ x=\frac{-36\pm50.91}{2} \\ \\ \text{ Therefore} \\ x=\frac{-36+50.91}{2}=7.46 \\ or \\ x=\frac{-36-50.91}{2}=-43.46 \end{gathered}[/tex]
x = -43.46 doesn't have sense here, so the length of each side of the octagon is 7.46
