In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagonals of this polygon is
- 30
- 54
- 64
- None of Above
The Correct Option is B
Solution and Explanation
To solve the given problem, we start with the known formulas for a polygon with \( n \) sides:
- Sum of interior angles = \( (n - 2) \times 180 \)
- Sum of exterior angles = \( 360^\circ \)
The interior angle sum is also expressed as: \[ (2n - 4) \times 90 \] This is just another form of writing \( (n - 2) \times 180 \), since: \[ (n - 2) \times 180 = (2n - 4) \times 90 \]
We are told that the difference between the sum of interior angles and exterior angles is: \[ (2n - 4) \times 90 - 360 = 120n \]
Let's simplify the left-hand side: \[ (2n - 4) \times 90 - 360 = 180n - 360 - 360 = 180n - 720 \] So the equation becomes: \[ 180n - 720 = 120n \] Subtract \( 120n \) from both sides: \[ 60n - 720 = 0 \] \[ 60n = 720 \] \[ n = 12 \]
Now that we know the polygon has \( n = 12 \) sides, we can find the number of diagonals.
The formula for the number of diagonals in an \( n \)-sided polygon is: \[ \frac{n(n - 3)}{2} \] Substituting \( n = 12 \): \[ \frac{12 \times (12 - 3)}{2} = \frac{12 \times 9}{2} = \frac{108}{2} = 54 \]
Therefore, the number of diagonals is: \( \boxed{54} \)
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