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

Strategy:

We are given the ratio of an interior angle to an exterior angle of a regular polygon. From this information, it is necessary to find the number of sides, 'n', of the polygon and then check which of the given statements about 'n' is true.

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\(^{\circ}\)).

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.

The correct statement is that \(n^2 - 8^2\) is an odd natural number.