The total number of sides in two regular polygons is 13, and total number of diagonals is 25. how many sides are in each polygon? PLS ANSWER QUICK!
let x be the sides of one polygon
y be the sides of the other polygon
x + y = 13
y = 13 - x
then the formula for the number of polygons is n(n - 3) / 2
x(x - 3) / 2 + y(y - 3) / 2 = 25
x(x - 3) / 2 + ( 13 - x) ( 13 - x - 3) / 2 = 25
x( x - 3 ) / 2 + ( 13 - x) ( 10 - x) / 2 = 25
solve for x
x = 8
y = 13 - 8 = 5
I dunno
The total number of sides in two regular polygons is 13, and total number of diagonals is 25. how many sides are in each polygon?
Number of sides of first polygon : x
number of sides in the 2nd = 13-x
number of diagonals of an n-gon = C(n,2) - n = n(n-1)/2 - n
(I subtracted n because the n sides are not diagonals)
number of diagonals for the x-gon
= C(x,2) - x = x(x-1)/2 - x
number of diagonals or the other
= C(13-x, 2) - (13-x) = (13-x)(12-x)/2 - 13 + x
(13-x)(12-x)/2 - 13 + x + x(x-1)/2 - x = 25
multiply and expand at the same time
156 - 25x + x^2 - 26 + 2x + x^2 - x - 2x = 50
2x^2 - 26x + 80 = 0
x^2 - 13x + 40 = 0
(x-5)(x-8) = 0
x = 5 or x = 8
Symmetric solution, that is,
if x = 5, 13-5 = 8
if x = 8, 13-8 = 5
So one is a pentagon, the other an octagon.
To find the number of sides in each polygon, we need to solve a system of equations based on the given information.
Let's assume the number of sides in the first polygon is "n", and the number of sides in the second polygon is "m".
1. Total number of sides: The total number of sides in two polygons is given as 13. So, our first equation is:
n + m = 13
2. Total number of diagonals: The number of diagonals in a polygon with "n" sides is given by the formula: D = (n * (n - 3)) / 2. Therefore, for the first polygon, the number of diagonals is [(n * (n - 3)) / 2]. Similarly, for the second polygon, the number of diagonals is [(m * (m - 3)) / 2]. The total number of diagonals is given as 25. So, our second equation is:
[(n * (n - 3)) / 2] + [(m * (m - 3)) / 2] = 25
Now, we have a system of two equations with two variables. To solve it, we can use substitution or elimination method.
Considering the urgency, let's solve the system using the elimination method:
1. Multiply the first equation by 2:
2(n + m) = 2(13)
2n + 2m = 26
2. Expand the second equation:
[(n^2 - 3n) + (m^2 - 3m)] / 2 = 25
n^2 + m^2 - 3n - 3m = 50
3. Rearrange the equations:
2n + 2m = 26 ---> 2n - 2m = 52 (multiply the equation by -1)
n^2 + m^2 - 3n - 3m = 50
4. Add the equations together:
(2n + 2m) + (n^2 + m^2 - 3n - 3m) = 26 + 50
n^2 + 2n - 3n + m^2 - 3m + 2m = 76
n^2 - n + m^2 - m = 76
Now we have one equation in terms of a single variable. Since the equation is a quadratic equation, we'll need to solve it:
n^2 - n + m^2 - m = 76
To solve this, we need additional information or constraints. Without further information or constraints, the solution for the values of "n" and "m" cannot be determined uniquely.
Please provide additional information or constraints to find a specific solution.