Question

A square, whose side is $2$ metres, has its corners cut away so as to form an octagon with all sides equal. Then find the length of each side of the octagon, in metres

• A. $\frac{2}{\sqrt{2}+1}$
• B. $\frac{\sqrt{2}}{\sqrt{2}+1}$
• C. $\frac{\sqrt{2}}{3}$
  
• D. none

Answer 1

Answer: (A)

$\frac{2}{\sqrt{2}+1}$

Let the length of AE be x as shown in the figure

In$\Delta AGE$

Using pythagora theorem

$G{E}^2=A{E}^2+A{G}^2=x^2+x^2\to GE=\sqrt{2}x$

Also AD=AE+EF+FD

$2=x+\sqrt{2}x+x\to x(2+\sqrt{2})=2\to \sqrt{2}x=\frac{2}{\sqrt{2}+1}\to GE=\frac{2}{\sqrt{2}+1}$

Length of side of octagon=$\frac{2}{\sqrt{2}+1}$