1957 AMC 12 Problem 31

Question

A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square. If the square has sides of one unit, the leg of each of the triangles has length:

\dfrac{2+\sqrt2}{3}

\dfrac{2-\sqrt2}{2}

\dfrac{1+\sqrt2}{2}

\dfrac{1+\sqrt2}{3}

\dfrac{2-\sqrt2}{3}

Answer: B

Concepts:regular polygon special right triangle radical

Difficulty rating: 1630

Small hint:

If each cut-off leg is

x,

an uncut horizontal octagon side has length

1-2x

Big hint:

A slanted octagon side is the hypotenuse

x\sqrt2

, and regularity makes these equal

Solution:

Let

x

be a leg of each corner triangle. The horizontal and vertical octagon sides have length

1-2x,

while the four slanted sides have length

x\sqrt2.

Thus

1-2x=x\sqrt2,

so

x=\frac1{2+\sqrt2}=\frac{2-\sqrt2}{2}.

Therefore, the correct answer is B.