Two sides of similar regular polygons are in the ratio of 5 : 4. The sum of their perimeters is 20. Find the
perimeters of each polygon
With X = the smaller perimeter,
X + (5/4)X = 20
Solve for X
To find the perimeters of each polygon, let's first assign variables to the lengths of the sides. Let's say the length of a side of the first polygon is 5x, and the length of a side of the second polygon is 4x.
The sum of their perimeters is 20, so we can set up an equation:
Perimeter of first polygon + Perimeter of second polygon = 20
Since the perimeters of regular polygons are given by the formula P = n * s, where P is the perimeter, n is the number of sides, and s is the length of a side, we can write the equation as:
(5x * n1) + (4x * n2) = 20
Now, we need to determine the values of n1 and n2 - the number of sides of each polygon. Since both polygons are regular, they will have the same number of sides. So let's denote the number of sides of each polygon as just n.
(5x * n) + (4x * n) = 20
Combining like terms:
9x * n = 20
Now, since n represents the number of sides of the polygon, it must be a positive integer. Looking at the equation, we see that if n = 1, then the product 9x * n = 9x, which is less than 20. If n = 2, then the product 9x * n = 18x, which is still less than 20. However, if n = 3, then the product 9x * n = 27x, which is greater than 20. This suggests that n should be 2.
Substituting n = 2 into the equation:
9x * 2 = 20
18x = 20
Dividing both sides by 18:
x = 20/18
Simplifying:
x = 10/9
Now we can find the lengths of the sides:
Length of a side of the first polygon = 5x = 5 * (10/9) = 50/9
Length of a side of the second polygon = 4x = 4 * (10/9) = 40/9
To find the perimeters of each polygon, we multiply the length of a side by the number of sides:
Perimeter of the first polygon = (50/9) * 2 = 100/9
Perimeter of the second polygon = (40/9) * 2 = 80/9
Therefore, the perimeters of the polygons are 100/9 and 80/9 respectively.