Respuesta :

JcAlmighty
The answer is 5 sides and 8 sides.

x - number of sides in one polygon
y - number of sides in another polygon
The total number of sides in two regular polygons is 13:
x + y = 13

The number of diagonals of a polygon with n sides is: (n - 3)*n/2 = (n²-3n)/2. 
Total number of diagonals is 25:
(x²-3x)/2 + (y²-3y)/2 = 25
(x² - 3x + y² - 3y)/2 = 25
x² - 3x + y² - 3y = 25 * 2
x² - 3x + y² - 3y = 50

We have system of two equations:
x + y = 13
x² - 3x + y² - 3y = 50
_____
x = 13 - y
x² - 3x + y² - 3y = 50
_____
Substitute x in the first equation:
(13 - y)² - 3(13 - y) + y² - 3y = 50
169 - 26y + y² - 39 + 3y + y² - 3y = 50
169 - 39 - 26y + 3y - 3y + y² + y² = 50
130 - 26y + 2y² = 50
2y² - 26y + 130 - 50 = 0
2y² - 26y + 80 = 0

Divide both sides by 2:
y² - 13y + 40 = 0
y² - 8y - 5y + 40 = 0
y*y - 8*y - 5 * y - 5 * (-8) = 0
y*(y-8) - 5(y-8) = 0
(y - 8)(y - 5) = 0
y - 8 = 0 or y - 5 = 0
y = 8 or y = 5

When y = 8, x = 13 - 8 = 5
When y = 5, x = 13 - 5 = 8
aksnkj

The figure that contains n number of sides in a regular manner is entitled to the polygon. The number of sides in polygons is 5 and 8 units.

Given-

The sum of the sides of the given polygons is 13 units.

The sum of the diagonals of the given polygons is 25 units.

Suppose, the first polygon has n sides and the second polygon has m sides. Now by the given condition from the question, we get,

[tex]n+m=13[/tex]

[tex]n=13-m[/tex]

This is our equation 1.

The formulated way to represent any polygon that has a k number of sides is mentioned below.

[tex]\rm{Diagonal\;in\;polygon\;that\;has \;k\;number\;of\;sides}=\dfrac{k(k-3)}{2}[/tex]

Using the above formula and given condition we can make another equation,

[tex]\dfrac{n(n-3)}{2}+\dfrac{m(m-3)}{2}=25[/tex]

How to solve the above equations?

Putting the value of n from in equation 1 into the above equation we get,

[tex]\dfrac{(13-m)(13-m-3)}{2}+\dfrac{m(m-3)}{2}=25[/tex]

[tex]\dfrac{(13-m)(10-m)}{2}+\dfrac{m^2-3m}{2}=25[/tex]

[tex]\dfrac{m^2-23m+130}{2}+\dfrac{m^2-3m}{2}=25[/tex]

[tex]{m^2-23m+130}+{m^2-3m}=25\times2[/tex]

[tex]2m^2-26m+130-50=0[/tex]

[tex]2m^2-26m+80=0[/tex]

Divide by two and solve it further,

[tex]m^2-13m+40=0[/tex]

[tex]m^2-8m-5m+40=0[/tex]

[tex]m(m-8)-5(m-8)=0[/tex]

[tex](m-8)(m-5)=0[/tex]

m=8 or m=5

When m=8,

n=13-8

n=5

When m=5

n=13-5

n=8

Hence, the number of sides in polygons is 5 and 8.

For more about the polygon, follow the link below-

https://brainly.com/question/19495361

Otras preguntas