Problem 256 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Given that the area of a polygon of \(n\) sides circumscribed about a circle is to the area of the circumscribed polygon of \(2 n\) sides as \(3: 2\), find \(n\).

Step-by-Step Solution

Answer

The value of 'n' that satisfies the given conditions is \(n = 6\)

Step 1: Setup the ratio

Set up the following ratio based on the given relationship between the areas of the two polygons: \(\frac{A_{n}}{A_{2n}} = \frac{3}{2}\)

Step 2: Insert appropriate proportionality

Knowing that the area of n-sided polygon and 2n-sided polygon circumscribed around a circle is directly proportional to their number of sides, rewrite the last relation as \(\frac{n}{2n} = \frac{3}{2}\)

Step 3: Simplify and Solve for n

Solve the simplified ratio by cross multiplying, it simplifies to \(n = 2*3 = 6\)

Key Concepts

Area of Polygon

The area of a polygon refers to the amount of space enclosed within its sides. For regular polygons, which are shapes with equal sides and angles, the area can be calculated using various formulas depending on the known values. One common way is via the formula: \[ A = \frac{1}{4} n s^2 \cot \frac{\pi}{n} \]where:

• \( A \) represents the area,
• \( n \) is the number of sides,
• \( s \) is the length of a side.

Understanding the area is crucial, as it's a fundamental aspect often needed to determine how shapes fit together, particularly in geometry problems involving ratios or comparisons as seen in this exercise.
The exercise hints at an interesting aspect of how certain properties of polygons relate directly to relationships such as ratios.

Circumscribed Circle

A circumscribed circle, sometimes called the circumcircle, is a circle that passes through all the vertices of a polygon. It is especially relevant in regular polygons, where the circumcircle allows all vertices to be equidistant from the circle's center.
Consider how a polygon with fewer sides may have a larger circumscribed circle compared to a polygon with more sides. As the number of sides increases, the shape of the polygon becomes more circular, and thus its circumscribed circle tends to fit more snugly around the figure.
In this exercise, the polygons being compared (one with \( n \) sides and another with \( 2n \) sides) are both circumscribed about the same circle. Their areas are proportionally related, showing how the circumscription affects other geometric properties, such as the relationship between their areas.

Number of Sides of Polygon

The number of sides in a polygon, denoted as \( n \), defines its structure and significantly influences its properties and geometrical relationships. In a polygon, the greater the number of sides, the closer it resembles a circle in shape, which has infinite sides.
The exercise uses the number of sides to explore a particular geometric relationship between polygons. When a polygon has \( n \) sides, and another similar polygon has \( 2n \) sides, various properties, such as area, change according to specific mathematical relationships.
In the given exercise, identifying \( n \) was crucial to solving the ratio between the two areas of circumscribed polygons, which was found to be \( 6 \). This highlights the role of \( n \) as a central factor in determining other related properties, or values within a geometric shape.