A square measures 18feet 6inches in length. If an octagon is inscribed in the square, then how long should each side of the octagon be?
Look at one of the corner triangles.
Let each side of the octogon be x
so the hypotenuse of that triangle is x
let the other two sides be y each
y^2 + y^2 = x^2
2y^2 = x^2
√2 y = x
y = x/√2
then x/√2 + x + x/√2 = 18.5
multiply each term by √2
x + √2 x + x = 18.5√2
x(1 + √2 + 1) = 18.5√2
x = 18.5√2/(2+√2) or appr. 7.66 ft or 7 feet 8 inches
thank you very much
To find the length of each side of the inscribed octagon, we need to determine the distance from the center of the square to one of its corners.
Given that the square measures 18 feet 6 inches in length, we can assume that it is a square with sides measuring 18 feet 6 inches.
To find the distance from the center of the square to one of its corners, we need to use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, one side of the right triangle is half the length of the square's side, and the other side is also half the length of the square's side. So, each side of the right triangle is (18 feet 6 inches) / 2 = 9 feet 3 inches.
Using the Pythagorean theorem, we can find the length of the hypotenuse, which represents the distance from the center of the square to one of its corners:
length of hypotenuse = √[(9 feet 3 inches)² + (9 feet 3 inches)²]
Converting the length to the same units for convenience, we get:
length of hypotenuse = √[(111 inches)² + (111 inches)²]
Simplifying further:
length of hypotenuse = √[12,321 + 12,321]
length of hypotenuse ≈ √24,642
length of hypotenuse ≈ 156.99 inches
Therefore, the length of each side of the octagon, which is equal to the distance from the center of the square to any of its corners, should be approximately 156.99 inches.