Yes, it is possible to construct a polygon whose sum of interior angles is 20 right angles. Let me show you the detailed process to determine how many sides this polygon has.
1. Understand the Problem:
We are given that the sum of the interior angles of a polygon is equal to 20 right angles. We need to determine if such a polygon exists and, if so, find out the number of sides this polygon has.
2. Convert Right Angles to Degrees:
A right angle is equivalent to 90 degrees. Therefore, 20 right angles can be converted to degrees as follows:
[tex]\[
20 \times 90 = 1800 \text{ degrees}
\][/tex]
3. Use the Polygon Interior Angle Sum Formula:
The sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by the formula:
[tex]\[
(n - 2) \times 180 \text{ degrees}
\][/tex]
We set this equal to the sum we calculated (1800 degrees):
[tex]\[
(n - 2) \times 180 = 1800
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
To find the number of sides [tex]\( n \)[/tex], we solve the equation for [tex]\( n \)[/tex]:
[tex]\[
n - 2 = \frac{1800}{180}
\][/tex]
[tex]\[
n - 2 = 10
\][/tex]
[tex]\[
n = 10 + 2
\][/tex]
[tex]\[
n = 12
\][/tex]
5. Conclusion:
Therefore, there exists a polygon whose sum of interior angles is 20 right angles, and the polygon has 12 sides.
So, the number of sides of the polygon is [tex]\( n = 12 \)[/tex].
