The interior angles of a polygon are in AP. The smallest angle is 120° and the common differences is 5". Find the number of sides of the polygon.
To find the number of sides of the polygon, you can use the following formula for the sum of interior angles in a polygon:
Sum of Interior Angles = (n - 2) × 180°
Where "n" is the number of sides of the polygon.
In this case, you are given that the smallest angle is 120°, and the common difference between the interior angles is 5°. So, you can set up an arithmetic progression (AP) to find the sum of the interior angles:
First angle (a) = 120° Common difference (d) = 5°
The sum of an AP is given by:
Sum = n/2 * [2a + (n-1)d]
Now, plug in the values:
Sum = n/2 * [2(120°) + (n-1)(5°)]
Simplify:
Sum = n/2 * [240° + 5°(n-1)]
Sum = n/2 * [240° + 5n - 5°]
Sum = n/2 * [5n + 235°]
Now, you know that the sum of interior angles of the polygon is equal to (n - 2) × 180°:
n/2 * [5n + 235°] = (n - 2) × 180°
Now, solve for "n":
n[5n + 235°] = 2(n - 2) × 180°
5n^2 + 235n = 2(180n - 360)
5n^2 + 235n = 360n - 720
Rearrange:
5n^2 - 125n - 720 = 0
Now, you can solve this quadratic equation for "n." You can either use the quadratic formula or factor it. Factoring, you get:
(n - 9)(5n + 80) = 0
Setting each factor equal to zero:
- n - 9 = 0 => n = 9
- 5n + 80 = 0 => 5n = -80 => n = -16 (but this doesn't make sense in this context)
So, the number of sides of the polygon is 9.
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