the angle in one regular polygon is to that in another as 3:2 and the number of the sides in first is twice that in second determine the the number of sides of 2 polygons
let the number of sides of the second be n
then the number of sides of the first is 2n
size of angle of the second = 180(n-2)/n
size of angle of the first = 180(2n-2)/(2n)
but that ratio is 3:2
[180(n-2)/n] / [180(2n-2)/(2n)] = 2/3
after some canceling and cross-multiplying ...
n = 4
one polygon is a square , the other is an octogon
check:
(4 sides and 8 sides)
each angle in a square is 90°, each angle in an octogon is 135°
135 : 90 = 3 : 2
To determine the number of sides in each of the two polygons, we can set up an equation based on the given information.
Let's assume that the first polygon has "x" sides and the second polygon has "y" sides.
According to the given information, the ratio of the angles in the two polygons is 3:2. This means that each angle in the first polygon is three times the size of each angle in the second polygon.
The formula to calculate the interior angle of a regular polygon is:
Interior angle = (n-2) * 180 / n, where "n" represents the number of sides in the polygon.
Using this formula, we can set up the equation:
3 * [(x-2) * 180 / x] = 2 * [(y-2) * 180 / y]
Simplifying the equation further:
3(x-2) / x = 2(y-2) / y
Cross-multiplying:
3(x-2) * y = 2(x) * (y-2)
Expanding and rearranging the equation:
3xy - 6y = 2xy - 4x
Now, we know that the number of sides in the first polygon is twice that of the second polygon:
x = 2y
Substituting this value into our equation:
3(2y)(y) - 6y = 2(2y)(y) - 4(2y)
6y^2 - 6y = 4y^2 - 8y
Rearranging the equation:
6y^2 - 4y^2 - 6y + 8y = 0
2y^2 + 2y = 0
Factoring out "2y":
2y(y + 1) = 0
Setting each factor equal to zero:
2y = 0 or y + 1 = 0
Solving for "y":
y = 0/2 or y = -1
Since we cannot have a polygon with zero sides or a negative number of sides, we can discard these solutions.
Therefore, the number of sides in the second polygon is y = 0/2 = 0.
Knowing that the first polygon has twice the number of sides, the number of sides in the first polygon would be 2 * 0 = 0.
Thus, neither of the polygons has any sides.