Problem 70 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

For a regular polygon, let \(r\) and \(R\) be the respective radii of the inscribed and the circumscribed circles. A false statement among the following is (A) There is a regular polygon with \(\frac{r}{R}=\frac{1}{\sqrt{2}}\) (B) There is a regular polygon with \(\frac{r}{R}=\frac{2}{3}\) (C) There is a regular polygon with \(\frac{r}{R}=\frac{\sqrt{3}}{2}\) (D) There is a regular polygon with \(\frac{r}{R}=\frac{1}{2}\)

Step-by-Step Solution

Answer

Option B is the false statement.

Step 1: Understanding the Relationship between r and R

For a regular polygon with n sides, the ratio \( \frac{r}{R} \) is given by \( \cos \left( \frac{\pi}{n} \right) \). We will use this relationship to evaluate each option.

Step 2: Analyzing Option A

For \( \frac{r}{R} = \frac{1}{\sqrt{2}} \), we set \( \cos \left( \frac{\pi}{n} \right) = \frac{1}{\sqrt{2}} \). The value \( \frac{1}{\sqrt{2}} \) corresponds to \( \pi/4 \), therefore, \( \frac{\pi}{n} = \frac{\pi}{4} \), which means \( n = 4 \). This is true because a regular polygon (square) exists for \( n = 4 \).

Step 3: Analyzing Option B

For \( \frac{r}{R} = \frac{2}{3} \), solve \( \cos \left( \frac{\pi}{n} \right) = \frac{2}{3} \). There's no simple substitution here, but using numerical or graphical methods, we find that a regular polygon with \( \cos \left( \frac{\pi}{n} \right) = \frac{2}{3} \) exists approximately when \( n = 6.464 \, (rounded) \). Since \( n \) must be an integer, this is false.

Step 4: Analyzing Option C

For \( \frac{r}{R} = \frac{\sqrt{3}}{2} \), this corresponds to \( \cos \left( \frac{\pi}{n} \right) = \frac{\sqrt{3}}{2} \) for \( \pi/6 \) angles. Therefore, \( \frac{\pi}{n} = \frac{\pi}{6} \), leading to \( n = 6 \). A regular hexagon fulfills this criterion perfectly.

Step 5: Analyzing Option D

Here, \( \frac{r}{R} = \frac{1}{2} \) implies \( \cos \left( \frac{\pi}{n} \right) = \frac{1}{2} \). This corresponds to \( \pi/3 \), thus \( \frac{\pi}{n} = \frac{\pi}{3} \), making \( n = 3 \). This corresponds to an equilateral triangle, which is a known occurrence.

Key Concepts

Inscribed Circle

An inscribed circle of a regular polygon is the largest possible circle that fits entirely within the polygon such that each of its sides is tangent to the circle. The center of this inscribed circle, also known as the incenter, is equidistant from all sides of the polygon. This property ensures that the radius of the inscribed circle, often denoted as \( r \), can be related to the structure of the polygon itself.
The radius \( r \) is instrumental when calculating the area or other properties of the polygon, especially in connection with the polygon's circumcircle. Each side of the polygon touches (is tangent to) the inscribed circle at exactly one point. This tangency directly relates the radius to the cosine of specific angles within these polygons.

Circumscribed Circle

A circumscribed circle, or circumcircle, of a regular polygon is a circle that passes through all vertices of the polygon. Each vertex of the regular polygon lies on the circumcircle, making the circumcircle centered at the same point as the inscribed circle, known as the circumcenter. The radius of the circumcircle is denoted by \( R \).
This concept is important because a circumcircle helps in finding relationships between the polygon's geometry and trigonometric properties. In regular polygons, understanding how the circumradius \( R \) and the number of sides \( n \) affect each other gives rise to critical mathematical insights, such as the formula relating \( r \), \( R \), and the cosine of internal angles.

Cosine Ratio

The cosine ratio is a key trigonometric function used to relate the inscribed and circumscribed circles in the context of regular polygons. For a regular polygon with \( n \) sides, the ratio \( \frac{r}{R} \) is equivalent to the cosine of an internal angle, specifically \( \cos \left( \frac{\pi}{n} \right) \).
This relationship is crucial when determining the properties of the polygon based on the angles formed by its sides and the circle's central points. Each cosine value corresponds to a specific polygon, with familiar angles like \( \pi/4 \), \( \pi/6 \), and \( \pi/3 \) matching with squares, hexagons, and equilateral triangles, respectively. Such relationships allow mathematicians to predict possible regular polygons with certain \( \frac{r}{R} \) ratios, enhancing our understanding of polygonal symmetries and their respective geometric properties.