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.

1957 AMC 12 Problem 31

Answer: B

Difficulty rating: 1630

Small Hint:

Big Hint:

Solution:

Problem 31 in Other Years