Answer :
Final answer:
To find the value of n, the number of sides of a regular polygon, given the angle GAF in a geometry problem involving a regular hexagon, we can utilize the internal angle formula for regular polygons. Through a series of calculations, we determine that the polygon in question is an octagon, hence n equals 8.
Explanation:
The question asked is related to the properties of a regular polygon and internal angles. Specifically, the question involves finding the value of n, the number of sides of a regular polygon, given that one side of a regular hexagon, which is a six-sided polygon, forms the side of this regular polygon and that the angle GAF is 105°.
To solve this, we will use the formula for calculating the internal angles of a regular polygon, which is (n-2)×180°/n where n is the number of sides of the polygon. Since we have a hexagon, we know the internal angle for a hexagon is 120°. For the angle GAF to be 105°, the side A-F must be a side of another regular polygon, which we are trying to determine.
Let's denote the internal angle of the unknown polygon as Int. Since the sum of the angles at a point is 360°, and considering that GAF is part of a regular polygon which meets with the hexagon at point A, we can express as 360 - Int - 120 = 105°. Solving for Int gives us 135°. We can then use the regular polygon internal angle formula set equal to 135° and solve for n. This gives us (n-2)×180°/n = 135°. After solving, we find that n = 8, indicating that the polygon is an octagon.
Answer:
n = 8.
Step-by-step explanation:
The interior angle of a regular hexagon = 120 degrees.
So m < GAB = 360 - 120 - 105
= 135 degrees so the regular polygon has a total of 135n degrees.
Using the formula for the sum of interior angles of a polygon,
For the polygon with n sides:
Sum of interior angles = 180(n - 2)
135n = 180(n - 2)
135n = 180n - 360
45n = 360
n = 360/45 = 8.
