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One side of a regular hexagon ABCDEF forms the side of a regular polygon with n sides.

Angle GAF = 105°. Work out the value of n. ​

One side of a regular hexagon ABCDEF forms the side of a regular polygon with n sides Angle GAF 105 Work out the value of

Answer :

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.

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Rewritten by : Barada

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.