Question 942759: the ratio between the number of sides of two regular polygon is 3:4 and the ratio between the sum of their interior angles is 2:3.find the number of sides in each polygon.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let s1 = the sum of the angles of the first polygon.
let s2 = the sum of the angles of the second polygon.
let n1 = the number of sides of the first polygon.
let n2 = the number of sides of the second polygon.
the sum of the angles of a polygon is equal to 180 * (n-2)
for the first polygon, you get:
s1 = 180 * (n1 - 2)
for the second polygon, you get:
s2 = 180 * (n2 - 2)
the ratio of the sum of the angles of the polygons is 2/3.
this means that s1/s2 = 2/3
you can solve for s1 in this equation to get s1 = (2 * s2) / 3.
since s1 is equal to 180 * (n1 - 2), and since s2 is equal to 180 * (n2 - 2), you can substitute in the equation for s1 to get:
s1 = (2 * s2) / 3 becomes:
180 * (n1 - 2) = (2 * 180 * (n2 - 2)) / 3.
divide both sides of this equation by 180 to get:
(n1 - 2) = (2 * 180 * (n2 - 2)) / (3 * 180).
since 180 in the numerator and denominator cancel out, the equation becomes:
(n1 - 2) = (2 * (n2 - 2)) / 3.
add 2 to both sides of this equation to get:
n1 = (2 * (n2 - 2)) / 3 + 2.
since 2 is equal to 6/3, the equation becomes:
n1 = (2 * (n2 - 2)) / 3 + 6/3.
you can now consolidate under the common denominator to get:
n1 = (2 * (n2 - 2) + 6) /3.
simplify by expanding the distribution to get:
n1 = (2 * n2 - 4 + 6) / 3.
combine like terms to get:
n1 = (2 * n2 + 2) / 3.
you are given that the ratio of n1/n2 = 3/4
from that ratio, you can solve for n1 to get:
n1 = (3 * n2) / 4
you can now replace n2 with its equivalent to get:
n1 = (2 * n2 + 2) / 3 becomes:
(3 * n2) / 4 = (2 * n2 + 2) / 3.
multiply both sides of this equation by 12 to get:
3 * (3 * n2) = (2 * n2 + 2) * 4.
simplify to get:
9 * n2 = 8 * n2 + 8
subtract 8 * n2 from both sides of this equation to get:
n2 = 8
since the ratio of n1 / n2 = 3/4, then the equation of:
n1 / n2 = 3/4 becomes:
n1 / 8 = 3/4
multiply both sides of this equation by 8 to get:
n1 = (3 * 8) / 4 which becomes:
n1 = 24 / 4 which becomes:
n1 = 6
you now have:
n1 = 6
n2 = 8
the ratio of n1/n2 is equal to 6/8 which is equal to 3/4.
s1 = 180 * (n1 - 2) = 180 * (6 - 2) = 180 * 4 = 720
s2 = 180 * (n2 - 2) = 180 * (8 - 2) = 180 * 6 = 1080
the ratio of s1/s2 is equal to 720/1080 which is equal to 2/3.
the ratio of the number of sides of the two polygons is equal to 3/4 as indicated in the problem statement.
the ratio of the sum of the angles of the two polygons is equal to 2/3 as indicated in the problem statement.
the solution is confirmed as good.
the solution to the problem is that the smaller polygon has 6 sides and the larger polygon has 8 sides.
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