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

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 [2010] (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 false; no polygon has \( \frac{r}{R} = \frac{2}{3} \).

Step 1: Understanding the Polygons and Their Properties

In a regular polygon with side length \( s \) and \( n \) sides, there are two important circles: the inscribed (incircle) and circumscribed (circumcircle) circles. \( r \) is the radius of the incircle, and \( R \) is the radius of the circumcircle.

Step 2: Determine the General Formula for \( \frac{r}{R} \)

The ratio of the inradius \( r \) to the circumradius \( R \) is determined by the formula:\[ \frac{r}{R} = \cos\left( \frac{\pi}{n} \right) \] where \( n \) is the number of sides of the polygon.

Step 3: Test Option (A) Ratio \( \frac{r}{R} = \frac{1}{\sqrt{2}} \)

To check if such a polygon exists, set \( \cos\left( \frac{\pi}{n} \right) = \frac{1}{\sqrt{2}} \). Solving, we find \( \frac{\pi}{n} = \frac{\pi}{4} \), giving \( n = 4 \). A square exists and satisfies this condition, so this statement is true.

Step 4: Test Option (B) Ratio \( \frac{r}{R} = \frac{2}{3} \)

Set \( \cos\left( \frac{\pi}{n} \right) = \frac{2}{3} \). This implies solving \( \cos(x) = \frac{2}{3} \), which corresponds to an angle that doesn't typically align with a regular polygon angle. Hence, a polygon with \( r/R = 2/3 \) does not exist, making this statement false.

Step 5: Test Option (C) Ratio \( \frac{r}{R} = \frac{\sqrt{3}}{2} \)

Set \( \cos\left( \frac{\pi}{n} \right) = \frac{\sqrt{3}}{2} \). Solving gives \( \frac{\pi}{n} = \frac{\pi}{6} \), resulting in \( n = 6 \), which is a regular hexagon. Therefore, this statement is true.

Step 6: Test Option (D) Ratio \( \frac{r}{R} = \frac{1}{2} \)

Set \( \cos\left( \frac{\pi}{n} \right) = \frac{1}{2} \). Solving gives \( \frac{\pi}{n} = \frac{\pi}{3} \), resulting in \( n = 3 \), which is an equilateral triangle. Therefore, this statement is true.

Key Concepts

Ratio of Inradius to Circumradius

In the study of regular polygons, understanding the ratio of the inradius (\( r \)) to the circumradius (\( R \)) is essential. Every regular polygon has two unique circles associated with it:

• The inradius (\( r \)): The radius of the circle inside the polygon that touches all its sides.
• The circumradius (\( R \)): The radius of the circle outside the polygon that passes through all its vertices.

This ratio allows us to gain insights into the geometric structure of the polygon. The formula for the ratio of the inradius to the circumradius is given by:\[\frac{r}{R} = \cos\left( \frac{\pi}{n} \right)\]where \( n \) is the number of sides of the polygon. This relationship highlights how cosine is directly involved in expressing this ratio in terms of the polygon's sides. For instance, in a square (\( n = 4 \)), this ratio is \( \frac{1}{\sqrt{2}} \), illustrating the role of cosine in determining geometric properties.

Polygon Properties

Regular polygons have unique characteristics that influence their geometric relationships. They are equilateral (all sides are equal) and equiangular (all angles are equal). This uniformity makes analyzing their properties relatively straightforward.

• Number of sides \( n \): Determines the poly's shape, be it a triangle (3 sides) or hexagon (6 sides).
• Angles: The internal angles are exceedingly well-ordered for computation based on the number of sides. This gives rise to predictable patterns for trigonometric relationships like cosine.
• Symmetry: Regular polygons exhibit a high degree of symmetry, which simplifies the understanding of their geometric placements, like how incircles and circumcircles fit.

These properties influence essential mathematical calculations, such as determining the inradius and circumradius. Understanding the properties allows students to grasp more complex geometric concepts and analyze different polygon configurations.

Cosine and Regular Polygons

Cosine is a trigonometric function that plays a pivotal role in the mathematics of regular polygons. It appears in the relationship between the inradius and circumradius of these polygons. The cosine function helps us to gender the ratio \( \frac{r}{R} = \cos\left( \frac{\pi}{n} \right) \) where \( n \) represents the number of sides. Here are key aspects of how cosine functions in regular polygons:

• Angle Measurement: It helps find precise measures like \( \cos\left( \frac{\pi}{n} \right) \) for calculating the circle radii involved.
• Trigonometric Values: Cosine values correspond to specific internal angles at polygon vertices, giving a direct link to geometric proportions.
• Solving Polygon Problems: By equating a specific cosine value to potential \( \frac{r}{R} \) ratios, you can deduce the feasibility of certain geometric forms.

Ultimately, cosine provides a bridge linking pure trigonometric functions to tangible geometric properties within the realm of regular polygons, revealing more about the shapes and their circular interactions.