Similar ratio and ratio perimeter for two regular octagons with area=18in squared and 50in squared.
9:25, 3:5
or
9:25, 9:25
To find the ratio of the side lengths of the two regular octagons, we need to find the square root of the ratio of their areas.
Let's denote the side lengths of the first octagon as "a" and the side lengths of the second octagon as "b".
For the first octagon with an area of 18 square inches, we can use the formula for the area of a regular polygon to get:
Area of the first octagon = (2 * a^2) * (1 + √2) = 18
Simplifying the equation, we have:
a^2 * (1 + √2) = 9
Next, we can solve for "a" by dividing both sides of the equation by (1 + √2):
a^2 = 9 / (1 + √2)
To find "a", we take the square root of both sides:
a = √(9 / (1 + √2))
Using the same approach, we can find the side length "b" for the second octagon with an area of 50 square inches:
b = √(50 / (1 + √2))
Now, we can determine the ratio of the side lengths by dividing "a" by "b":
Ratio of Side Lengths = a / b
So, to calculate the ratio, substitute the values of "a" and "b" into the equation and simplify:
Ratio of Side Lengths = (√(9 / (1 + √2))) / (√(50 / (1 + √2)))
Simplifying this expression would give you the ratio of the side lengths for the two octagons.
To determine the ratio of the perimeters, we can now multiply the ratio of the side lengths by the ratio of the number of sides (8 sides for both octagons):
Ratio of Perimeters = Ratio of Side Lengths * (Number of Sides)
So, if the ratio of side lengths is 9:25, the ratio of the perimeters would be 9 * 8 : 25 * 8, which simplifies to 72:200 or can be further simplified to 9:25.
Therefore, the correct answer is 9:25 for the ratio of side lengths and 9:25 for the ratio of perimeters.