Question

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

(a) \(\frac{\sqrt2}{\sqrt2+1}\)

(b) \(\frac{2}{\sqrt2+1}\)

(c) \(\frac{2}{\sqrt2-1}\)

(d) \(\frac{\sqrt2}{\sqrt2-1}\)

Answer 1

(a) \(\frac{\sqrt2}{\sqrt2+1}\)

Let the length of the edge cut away from the corners of the square be \(x\). Since the resulting figure is a regular octagon.

\(\sqrt{x^2+x^2}\) = 2 - 2\(x\)  ⇒ \(\sqrt2x\) = 2 - 2\(x\)

⇒ \(\sqrt2x\) +  2\(x\) = 2  ⇒ \(\sqrt2x\)(1+\(\sqrt2\)) = 2

⇒ \(x\) = \(\frac{2}{\sqrt2(1+\sqrt2)}=\frac{\sqrt2}{1+\sqrt2}\)