Question

An artist has been commissioned to make a stained glass window in the shape of a regular octagon. The octagon must fit inside an 20-in. square space. Determine the length of each side of the octagon. Round to the nearest hundredth of an inch

Answer 1

To determine the length of each side of the octagon, use the Pythagorean theorem to find the length of one of its diagonals. The diagonal divides the octagon into two congruent right triangles. The length of each side of the octagon is approximately 14.14 inches.

Step-by-step explanation:

To determine the length of each side of the octagon, we need to find the length of one of its diagonals. Since an octagon can be divided into four congruent isosceles triangles, we can use the Pythagorean theorem to find the length of the diagonal. Let's call the side length of the octagon 's' and the diagonal length 'd'. The diagonal divides the octagon into two congruent right triangles. The hypotenuse of each triangle is the diagonal, and the legs are two sides of the octagon.

Using the Pythagorean theorem, we have: s^2 + s^2 = d^2. Simplifying this equation, we get: 2s^2 = d^2. Taking the square root of both sides, we have: sqrt(2s^2) = sqrt(d^2). Simplifying further, we get: s * sqrt(2) = d. Since the octagon must fit inside a 20-inch square, the diagonal cannot be longer than 20 inches. Therefore, we have the inequality: d <= 20.

Substituting the expression for 'd' found earlier, we have: s * sqrt(2) <= 20. Dividing both sides of the inequality by sqrt(2), we get: s <= 20 / sqrt(2). Plugging this value into a calculator, we find that s <= 14.14 (rounded to the nearest hundredth). Therefore, the length of each side of the octagon must be 14.14 inches (rounded to the nearest hundredth).

Answer 2

Answer: 8 Inches

Step-by-step explanation:

First thing we do is to draw an octagon inscribed inside a square, as shown in the picture attached,

The we know one side of the square is 20 inches, and one side consist of the addition of a length of the octagon, and two sides for a right angle triangle, which has the smaller angles inside as 45, 45 degrees making those two sides equal to each other, So to find the value of x which is labeled as the side of the octagon, we adding the two sides of the right angle triangle with the side of the octagon to equate to 20 inches

Since a side of the square is 20inches

From the diagram, y is labeled as the two equal sides of the right angle triangle

So

y + x + y = 20 (1)

Using Pythagoras theorem for the right angle triangle

y2 + y2 = x2

2y2 = x2

y2 = (x2)/2

y = x/root(2)

So putting this in the previous equation (1)

x/root(2) + x + x/root(2) = 20

2x/root(2) + x = 20

2x/root(2) + xroot(2)/root(2) = 20

x(2 + root(2))/root(2) = 20

x(2 + root(2)) = 20(root(2))

x = (20(root(2)))/(2 + root(2))

= 28.2843/3.4142 = 8.2343

So therefore answer is approximately 8 inches