Question

An artist has been commissioned to make a stained glass window in the shape of a regular octagon. The octagon must fit inside aPlease check photo

Answer 1

Given:

Length of each side of the square = 12 in

Let's find the length of each side of the regular octagon.

We can see the parts of the rectangle which are not a part of the octagon for right triangles.

Since the two legs are equal, this means the triangle is a 45-45-90 degrees special right triangle.

Now, apply Pythagorean theorem:

$x^2+x^2=y^2$

Also, we know the length of the two sides plus one leg of the octagon equals length of one side of the square.

Now, we have the second equation:

$x+x+y=12$

Now, let's solve both equations simultaneously:

$\begin{gathered} x^2+x^2=y^2 \\ x+x+y=12 \end{gathered}$

Solving further, we have:

$\begin{gathered} 2x^2=y^2 \\ 2x+y=12 \end{gathered}$

In the first equation take the square root of both sides:

$\begin{gathered} \sqrt{2x^2}=\sqrt{y^2} \\ \\ x\sqrt{2}=y \\ \\ y=x\sqrt{2} \end{gathered}$

Now, substitute x√2 for y in the second equation:

$\begin{gathered} 2x+y=12 \\ \\ 2x+x\sqrt{2}=12 \end{gathered}$

Factor out x:

$x(2+\sqrt{2})=12$

Divide each term by (2+√2):

$\begin{gathered} \frac{x(2+\sqrt{2})}{2}=\frac{12}{2+\sqrt{2}} \\ \\ x=3.5 \end{gathered}$

Now, to find the length of reach side of the octagon, given that the length of the leg of the triangle is 3.5, apply Pythagorean theorem:

$\begin{gathered} y=\sqrt{3.5^2+3.5^2} \\ \\ y=\sqrt{12.25+12.25} \\ \\ y=\sqrt{24.50} \\ \\ y=4.9\approx5 \end{gathered}$

Therefore, each side of the octagon is approximately 5 in.

ANSWER:

5 in