Sure, let's solve the problem step-by-step:
1. Understand the given ratio: The ratio of the interior angle to the exterior angle of a regular polygon is given as 13:2.
2. Sum of interior and exterior angles: Remember that for any polygon, the sum of an interior angle and its corresponding exterior angle is always 180 degrees.
3. Express the ratios in terms of angles:
- Let the interior angle be [tex]\(13x\)[/tex].
- Let the exterior angle be [tex]\(2x\)[/tex].
4. Set up the equation using the sum of angles:
[tex]\[
13x + 2x = 180 \text{ degrees}
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
15x = 180 \text{ degrees}
\][/tex]
[tex]\[
x = \frac{180}{15}
\][/tex]
[tex]\[
x = 12
\][/tex]
6. Find the measures of the angles:
- The interior angle is [tex]\(13x = 13 \times 12 = 156\)[/tex] degrees.
- The exterior angle is [tex]\(2x = 2 \times 12 = 24\)[/tex] degrees.
7. Use the exterior angle to find the number of sides ([tex]\(n\)[/tex]):
[tex]\[
\frac{360 \text{ degrees}}{\text{exterior angle}} = n
\][/tex]
[tex]\[
n = \frac{360}{24}
\][/tex]
[tex]\[
n = 15
\][/tex]
Therefore, the value of [tex]\(n\)[/tex] is 15. The polygon is a 15-sided regular polygon.
