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Chapter 28: Problem 17

The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is [2003] (a) \(\frac{a}{4} \cot \left(\frac{\pi}{2 n}\right)\) (b) \(a \cot \left(\frac{\pi}{n}\right)\) (c) \(\frac{a}{2} \cot \left(\frac{\pi}{2 n}\right)\) (d) \(a \cot \left(\frac{\pi}{2 n}\right)\).

Short Answer

Expert verified
(d) \(a \cdot \cot \left(\frac{\pi}{2n}\right)\)

Step by step solution

01

Understanding the Problem

We are asked to find the sum of the radii of the incircle (inscribed circle) and circumcircle (circumscribed circle) of a regular n-sided polygon with side length \(a\). We need to compare this sum to the given options.
02

Formula for Radius of Incircle

The radius \(r\) of the incircle of a regular polygon with \(n\) sides, each of length \(a\), is given by \(r = \frac{a}{2} \cot \left(\frac{\pi}{n}\right)\).
03

Formula for Radius of Circumcircle

The radius \(R\) of the circumcircle of a regular polygon with \(n\) sides and side length \(a\) is \(R = \frac{a}{2} \csc \left(\frac{\pi}{n}\right)\).
04

Sum of the Radii

Calculate the sum of the radii of the incircle and the circumcircle:\[R + r = \frac{a}{2} \csc \left(\frac{\pi}{n}\right) + \frac{a}{2} \cot \left(\frac{\pi}{n}\right)\] We simplify this by finding a common expression.
05

Simplification of Terms

Use the trigonometric identity: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Substitute these into the sum expression:\[\frac{a}{2} \cdot \frac{1}{\sin \left(\frac{\pi}{n}\right)} + \frac{a}{2} \cdot \frac{\cos \left(\frac{\pi}{n}\right)}{\sin \left(\frac{\pi}{n}\right)}\]Combine the terms:\[\frac{a}{2} \cdot \frac{1 + \cos \left(\frac{\pi}{n}\right)}{\sin \left(\frac{\pi}{n}\right)} = \frac{a}{2} \cdot \frac{2 \cos^2 \left(\frac{\pi}{2n}\right)}{2 \sin \left(\frac{\pi}{2n}\right) \cos \left(\frac{\pi}{2n}\right)}\]This simplifies further to:\[a \cdot \cot \left(\frac{\pi}{2n}\right)\]
06

Choose the Correct Option

The simplified expression matches option (d), which is \(a \cdot \cot \left(\frac{\pi}{2n}\right)\). Therefore, the correct answer is option (d).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inscribed Circle
In geometry, an inscribed circle, or incircle, of a polygon is a circle located inside the polygon. It touches all sides of the polygon at exactly one point per side. This unique circle is tangent to each side, indicating that it fits perfectly within the polygon. The center of this circle is called the "incenter," which is also a key feature of this configuration.
  • The radius of the inscribed circle (also called the inradius) is an important parameter.
  • For a regular polygon with side length \(a\) and number of sides \(n\), the formula for the inradius \(r\) is \(r = \frac{a}{2} \cot \left(\frac{\pi}{n}\right)\).
Thus, understanding the concept of an inscribed circle helps in finding various properties, such as area and perimeter, of the polygon itself.
Circumscribed Circle
A circumscribed circle, or circumcircle, is a circle that passes through all the vertices of a polygon. Essentially, it is the smallest circle in which a polygon can be completely enclosed. The center of this circle is known as the "circumcenter."
  • The radius of the circumcircle is crucial in determining the polygon's geometrical properties.
  • For a regular polygon with side length \(a\) and number of sides \(n\), the circumradius \(R\) is given by \(R = \frac{a}{2} \csc \left(\frac{\pi}{n}\right)\).
The circumcircle is particularly significant in regular polygons where it provides insight into the symmetry and balanced nature of the shape.
Regular Polygon
A regular polygon is defined as a polygon with all sides and all angles equal. These polygons are highly symmetrical and therefore have interesting geometric properties.
  • Some common examples include equilateral triangles, squares, and regular pentagons.
  • In regular polygons, both the inscribed and circumscribed circles have unique relationships with the polygon.
For a regular polygon with \(n\) sides and side length \(a\), the sum of the radii of the inscribed and circumscribed circles can be expressed as \(a \cdot \cot \left(\frac{\pi}{2n}\right)\). This expression is derived using trigonometric identities and plays a key role in many geometric calculations related to regular polygons.

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Most popular questions from this chapter

\(\mathrm{ABCD}\) is a trapezium such that \(\mathrm{AB}\) and \(\mathrm{CD}\) are parallel and \(\mathrm{BC} \perp \mathrm{CD}\). If \(\angle \mathrm{ADB}=\theta, \mathrm{BC}=p\) and \(\mathrm{CD}=q\) then \(\mathrm{AB}\) is equal to: (a) \(\frac{\left(p^{2}+q^{2}\right) \sin \theta}{p \cos \theta+q \sin \theta}\) (b) \(\frac{p^{2}+q^{2} \cos \theta}{p \cos \theta+q \sin \theta}\) (c) \(\frac{p^{2}+q^{2}}{p^{2} \cos \theta+q^{2} \sin \theta}\) (d) \(\frac{\left(p^{2}+q^{2}\right) \sin \theta}{(p \cos \theta+q \sin \theta)^{2}}\)

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