Problem 63 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In a triangle \(A B C, \angle C=\frac{\pi}{2}\) If \(r\) is the inradius and \(R\) is the circumradius of the the triangle \(A B C\), then 2 ( \(r+\) \(R\) ) equals (A) \(b+c\) (B) \(a+b\) (C) \(a+b+c\) (D) \(c+a\)

Step-by-Step Solution

Answer

The answer is (B) \(a+b\).

Step 1: Recall Definitions

The inradius \( r \) of a triangle is the radius of the inscribed circle, and the circumradius \( R \) is the radius of the circle that passes through all three vertices of the triangle. In a right triangle, these have special relationships.

Step 2: Use Formula for Inradius in Right Triangle

In a right triangle, the inradius \( r \) can be expressed as \( r = \frac{a+b-c}{2} \), where \( a \), \( b \), and \( c \) are the side lengths with \( c \) as the hypotenuse.

Step 3: Use Formula for Circumradius in Right Triangle

In a right triangle, the circumradius \( R \) is given by \( R = \frac{c}{2} \), as the hypotenuse forms the diameter of the circumcircle.

Step 4: Combine Terms in Expression

We seek \( 2(r+R) \). Substitute the expressions for \( r \) and \( R \):\[2 \left( \frac{a+b-c}{2} + \frac{c}{2} \right) = (a+b-c + c) = a+b\]

Step 5: Compare with Options

The expression simplifies to \( a+b \), which matches option (B).

Key Concepts

Inradius of a Triangle

The inradius of a triangle refers to the radius of the circle that fits snugly inside the triangle, touching all three sides. This circle is known as the incircle. All triangles have an inradius, but the calculation varies depending on the type of triangle.
For a right triangle, the formula to calculate the inradius \( r \) is:

• \( r = \frac{a+b-c}{2} \)

Here, \( a \) and \( b \) are the lengths of the legs, and \( c \) is the hypotenuse of the triangle.
This formula highlights a key attribute: in a right triangle, the inradius depends on how the legs and the hypotenuse interact. The inradius formula simplifies calculations for problems involving the triangle's area or perimeter and is especially valuable when examining inscribed circles.

Circumradius of a Triangle

The circumradius of a triangle is the radius of the circumcircle, which is a circle that circumscribes the triangle. This means it passes through all three vertices of the triangle. Each triangle has a unique circumcircle, and the formula for the circumradius \( R \) depends on the triangle's properties.
In a right triangle, the circumradius is particularly simple:

• \( R = \frac{c}{2} \)

In this formula, \( c \) denotes the hypotenuse of the triangle. For right triangles, the circumcircle's diameter is equal to the hypotenuse, making it straightforward to compute \( R \). This simplification arises because the triangle fits perfectly inside a semicircle when the hypotenuse is aligned with the circle's diameter. This property is often useful not just in theoretical exercises but also in real-world applications such as engineering and architecture.

Right Triangle Properties

A right triangle is a triangle in which one angle measures exactly 90 degrees. This structure leads to unique and often simpler geometric properties when compared to other triangles. Understanding these properties is crucial when working with right triangles in triangle geometry.
Key properties include:

• The Pythagorean theorem: \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse.
• The inradius formula: \( r = \frac{a+b-c}{2} \)
• The circumradius formula: \( R = \frac{c}{2} \)

These formulas facilitate calculations involving the properties of right triangles, making it easier to solve problems related to angles, side lengths, and areas. Right triangles frequently appear in various fields, from mathematics and physics to different branches of engineering, and their distinctive attributes offer practical solutions in design, construction, and analysis.