Question

The ratio of an interior angle to its corresponding exterior angle of a regular polygon is 9 : 2. If number of sides in the polygon is n, then select the CORRECT option.

• A. \(n^2 - 10^3\) is an odd natural number
• B. n is an even natural number
• C. \(n^2 - 8^2\) is an odd natural number
• D. \(n^2 - n\) is an odd natural number

Hint:

Using the property I + E = 180\(^{\circ}\) is often faster. If I/E = a/b, then I = 180 * (a/(a+b)) and E = 180 * (b/(a+b)). Here, E = 180 * (2/(9+2)) = 360/11. Then n = 360/E gives n=11.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the ratio of an interior angle to an exterior angle of a regular polygon. From this information, we need to find the number of sides, 'n', of the polygon and then check which of the given statements about 'n' is true.
Step 2: Key Formula or Approach:
For a regular polygon with 'n' sides: 1. Each exterior angle (E) is given by \(E = \frac{360^{\circ}}{n}\). 2. Each interior angle (I) is given by \(I = 180^{\circ} - E = 180^{\circ} - \frac{360^{\circ}}{n}\). 3. The sum of an interior angle and its corresponding exterior angle is always 180\(^{\circ}\) (I + E = 180\(^{\circ}\)).
Step 3: Detailed Explanation:
Let the interior angle be I and the exterior angle be E. We are given the ratio \(\frac{I}{E} = \frac{9}{2}\), which means \(I = \frac{9}{2}E\).
Using the property that \(I + E = 180^{\circ}\): \[ \frac{9}{2}E + E = 180^{\circ} \] \[ (\frac{9}{2} + 1)E = 180^{\circ} \] \[ \frac{11}{2}E = 180^{\circ} \] \[ E = \frac{180^{\circ} \times 2}{11} = \frac{360^{\circ}}{11} \] Now, we use the formula for the exterior angle: \(E = \frac{360^{\circ}}{n}\). \[ \frac{360^{\circ}}{11} = \frac{360^{\circ}}{n} \] From this, we can conclude that \(n = 11\).
Now we must check the given options with n = 11:
(A) \(n^2 - 10^3 = 11^2 - 1000 = 121 - 1000 = -879\). This is not a natural number. So, (A) is incorrect.
(B) n is an even natural number. n=11, which is an odd number. So, (B) is incorrect.
(C) \(n^2 - 8^2 = 11^2 - 8^2 = 121 - 64 = 57\). 57 is an odd natural number. So, (C) is correct.
(D) \(n^2 - n = 11^2 - 11 = 121 - 11 = 110\). 110 is an even number. So, (D) is incorrect.
Step 4: Final Answer:
The correct statement is that \(n^2 - 8^2\) is an odd natural number.