To find the number of sides in a regular polygon where the exterior angle and the interior angle are in the ratio [tex]\(2:7\)[/tex], follow these steps:
1. Understand the Relationship Between Angles:
- Let the measure of the exterior angle be [tex]\(2x\)[/tex] degrees.
- Let the measure of the interior angle be [tex]\(7x\)[/tex] degrees.
2. Sum of Angles in a Polygon:
- For any polygon, the exterior angle and the interior angle at any vertex add up to [tex]\(180^\circ\)[/tex].
- Hence, [tex]\(2x + 7x = 180^\circ\)[/tex].
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
9x = 180
\][/tex]
[tex]\[
x = \frac{180}{9} = 20
\][/tex]
4. Determine the Measure of Each Angle:
- The exterior angle is [tex]\(2x = 2 \times 20 = 40^\circ\)[/tex].
- The interior angle is [tex]\(7x = 7 \times 20 = 140^\circ\)[/tex].
5. Use the Exterior Angle to Find the Number of Sides:
- The measure of an exterior angle of a regular polygon is given by [tex]\(\frac{360^\circ}{n}\)[/tex], where [tex]\(n\)[/tex] is the number of sides.
[tex]\[
40 = \frac{360}{n}
\][/tex]
6. Solve for [tex]\(n\)[/tex]:
[tex]\[
n = \frac{360}{40} = 9
\][/tex]
Therefore, the number of sides in the polygon is [tex]\(9\)[/tex].
