Word problems

Question

The corner of a square the length of whose side is 2 are cut off in such a regular octagon remains. What is the length of a side of this octagon?

Answer 1

I hope you have a diagram of your own. Your diagram should have 4 right triangles being cut off from each of the corners of the square.

 

Call each of the legs of these right triangles x. Using the Pythagorean theorem, the hypoetnuse is sqt 2x^2.

This hypotenuse must be the same as all the sides of the regular octagon.. The sides of the octagon that lie on the side of the square can be represented as 2 - 2x...look at the diagram.

 

So 2 - 2x = sqrt 2x^2 will result in a regular octagon. Square both sides to get

4 - 8x + 4x^2 = 2x^2    It's a quadratic equation, so bring all the terms to one side.

2x^2 - 8x + 4 = 0    Use the quadratic formula to get

 

x = (8 +- sqrt(64-4*2*4) ) / 4

x = (8+- sqrt32) / 4

x = 2 +- sqrt 2   or

 

2 + sqrt2 or 2- sqrt 2

 

2 + sqrt2 is bigger than the original side of the square and is therfore rejected.

2 - sqrt2 is what must be cut off.

 

So the side of the octagon is 2 - 2(2 - sqrt 2) = -2 + 2sqrt 2 which is approximately .8284 for each side of the octagon.

 

let's check by using the Pythagorean theorem on the little triangles to see if the hypotenuses are the same number. I'll use the fact that 

2 - sqrt 2 is approximately .586.

 

.586^2 + .586^2 = hyp^2

.686792    =    hyp^2

.8287      =    hyp

 

That's pretty darn close to the answer using the first method. So I'm convinced that the side of the regular octagon is

 

EXACTLY    -2 + 2sqrt 2     OR

APPROXIMATELY .8284.

 

Hope you understood this.