Question

the total number of sides in 2 regular polygons is 13, and the total number of diagonals is 25. how many sides are in each polygon

Answer 1

Let the number of sides of the two polygons be denoted as \( n_1 \) and \( n_2 \). According to the problem, we have the following two equations:

1. The total number of sides:
\[
n_1 + n_2 = 13
\]

2. The total number of diagonals:
The formula for the number of diagonals \( D \) in a polygon with \( n \) sides is given by:
\[
D = \frac{n(n-3)}{2}
\]
Therefore, the total number of diagonals in the two polygons can be expressed as:
\[
D_1 + D_2 = 25
\]
where \( D_1 = \frac{n_1(n_1 - 3)}{2} \) and \( D_2 = \frac{n_2(n_2 - 3)}{2} \). Thus, we rewrite the second equation as:
\[
\frac{n_1(n_1 - 3)}{2} + \frac{n_2(n_2 - 3)}{2} = 25
\]
Multiplying through by 2 to eliminate the fraction:
\[
n_1(n_1 - 3) + n_2(n_2 - 3) = 50
\]

Now we can express \( n_2 \) in terms of \( n_1 \):
\[
n_2 = 13 - n_1
\]

Substituting \( n_2 \) into the diagonal equation:
\[
n_1(n_1 - 3) + (13 - n_1)((13 - n_1) - 3) = 50
\]
This simplifies to:
\[
n_1(n_1 - 3) + (13 - n_1)(10 - n_1) = 50
\]

Expanding the second term:
\[
n_1(n_1 - 3) + (130 - 13n_1 - 10n_1 + n_1^2) = 50
\]
Combining like terms:
\[
n_1(n_1 - 3) + (n_1^2 - 23n_1 + 130) = 50
\]

Now simplifying further:
\[
n_1^2 - 3n_1 + n_1^2 - 23n_1 + 130 = 50
\]
Combining terms gives:
\[
2n_1^2 - 26n_1 + 130 = 50
\]
Subtracting 50 from both sides:
\[
2n_1^2 - 26n_1 + 80 = 0
\]

Dividing the entire equation by 2:
\[
n_1^2 - 13n_1 + 40 = 0
\]

Next, we factor this quadratic equation:
\[
(n_1 - 8)(n_1 - 5) = 0
\]
Thus, we have two possible solutions for \( n_1 \):
\[
n_1 = 8 \quad \text{or} \quad n_1 = 5
\]

If \( n_1 = 8 \), then:
\[
n_2 = 13 - 8 = 5
\]
If \( n_1 = 5 \), then:
\[
n_2 = 13 - 5 = 8
\]

Hence, the two polygons have \( n_1 = 8 \) sides and \( n_2 = 5 \) sides, respectively.

To confirm, we can calculate the number of diagonals:
- For a polygon with 8 sides:
\[
D_1 = \frac{8(8-3)}{2} = \frac{8 \times 5}{2} = 20
\]
- For a polygon with 5 sides:
\[
D_2 = \frac{5(5-3)}{2} = \frac{5 \times 2}{2} = 5
\]

The total number of diagonals is \( 20 + 5 = 25 \), which matches the given condition.

Thus, the number of sides in the two polygons is \( 8 \) and \( 5 \).