Problem 48 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If in a triangle, \(R\) and \(r\) are the circum radius and in radius, respectively, then the H.M. of the ex-radii of the triangle is (a) \(3 r\) (b) \(2 R\) (c) \(R+r\) (d) None

Step-by-Step Solution

Answer

The harmonic mean of the ex-radii of the triangle is \(R - r\), hence none of the given options (a), (b), (c) are correct.

Step 1: Understanding the Relations

The relationship between the circumradius \(R\), inradius \(r\), and ex-radius \(r_1, r_2, r_3\) of a triangle is such that \(\frac{1}{R} = \frac{1}{r} - \frac{1}{r_1} = \frac{1}{r} - \frac{1}{r_2} = \frac{1}{r} - \frac{1}{r_3}\) and \(R = r + r_k\) for \(k = 1,2,3\). Therefore, \(r_1 = r_2 = r_3 = R - r\). This means that all ex-radii are equal.

Step 2: Calculating Harmonic Mean

The Harmonic Mean (H.M.) of two or more quantities is the reciprocal of the arithmetic mean of their reciprocals. As all ex-radii are the same, we can use one of them to calculate H.M. Thus, H.M. is \(\frac{3}{(\frac{1}{r_1}+\frac{1}{r_1}+\frac{1}{r_1})} = \frac{3}{\frac{3}{r_1}} = r_1 = R - r\)

Step 3: Finalizing the Answer

Thus, the Harmonic Mean of the ex-radii of a triangle is \(R - r\) and hence the answer doesn't match any of the given options (a), (b), (c).

Key Concepts

Harmonic Mean

The Harmonic Mean is a useful concept in mathematics, especially when dealing with rates or ratios. It's different from the Arithmetic and Geometric Means. Instead of simply averaging numbers, the Harmonic Mean involves the reciprocals of the numbers.
In simple terms, if you have a set of values, the Harmonic Mean is calculated by dividing the number of values by the sum of the reciprocals of the values.
For example, for three values, the formula would be:

• Harmonic Mean = \( \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3}} \)

This method is often used when the values are rates, like speed or density. It's particularly helpful when the average of rates gives a more accurate picture than a simple arithmetic mean.

circumradius

The circumradius is an important element in triangle geometry. It represents the radius of the circle that passes through all three vertices of the triangle.
This circle is known as the circumcircle. Every triangle has a unique circumcircle, and the circumradius is denoted by \( R \).
In terms of calculation, if you know the sides \( a, b, c \) and the area \( A \) of the triangle, you can find the circumradius using the formula:

• \( R = \frac{abc}{4A} \)

The circumradius helps in connecting various triangle properties and relationships between the sides, angles, and other elements of the triangle. Understanding the circumradius can also lead to deeper insights into trigonometric identities and properties.

inradius

The inradius of a triangle is just as fascinating. It is the radius of the incircle, which is the circle that fits perfectly within the triangle, touching all three sides.
The inradius is denoted by \( r \), and it can be found using the formula:

• \( r = \frac{A}{s} \)

Here, \( A \) is the area of the triangle, and \( s \) is the semi-perimeter (half the perimeter) of the triangle.
The inradius plays a crucial role in several geometric properties and can be used to determine the area of the triangle. It also connects with the circumradius and ex-radii, allowing for relationships like those seen in various triangle inequalities and equalities.