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,
\(n+m=13\)
\(n=13-m\)
This is our equation 1.
The formulated way to represent any polygon that has a k number of sides is mentioned below.
\(\rm{Diagonal\;in\;polygon\;that\;has \;k\;number\;of\;sides}=\dfrac{k(k-3)}{2}\)
Using the above formula and given condition we can make another equation,
\(\dfrac{n(n-3)}{2}+\dfrac{m(m-3)}{2}=25\)
How to solve the above equations?
Putting the value of n from in equation 1 into the above equation we get,
\(\dfrac{(13-m)(13-m-3)}{2}+\dfrac{m(m-3)}{2}=25\)
\(\dfrac{(13-m)(10-m)}{2}+\dfrac{m^2-3m}{2}=25\)
\(\dfrac{m^2-23m+130}{2}+\dfrac{m^2-3m}{2}=25\)
\({m^2-23m+130}+{m^2-3m}=25\times2\)
\(2m^2-26m+130-50=0\)
\(2m^2-26m+80=0\)
Divide by two and solve it further,
\(m^2-13m+40=0\)
\(m^2-8m-5m+40=0\)
\(m(m-8)-5(m-8)=0\)
\((m-8)(m-5)=0\)
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
