Problem 146 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the circumradius of the equilateral triangle of side \(2 \sqrt{3} \mathrm{~cm}\).

Step-by-Step Solution

Answer

The circumradius of the equilateral triangle is 2 cm.

Step 1: Identifying the given data

The given data from the problem is that the side of the equilateral triangle, denoted by `a`, is \(2\sqrt{3}\mathrm{~cm}\). This is the key data that'll be used in the solution process.

Step 2: Using the formula for circumradius of an equilateral triangle

The formula for the circumradius 'R' of an equilateral triangle is given by R= a/√3. Substituting the value of 'a' from the problem into this formula gets you R = \(2\sqrt{3}\) / √3.

Step 3: Simplifying the expression

On simplifying the expression, \(2\sqrt{3}\) / √3, you get R = \(2\sqrt{3}\) * /√3 * √3/3, which simplifies further to R = 2 cm.

Key Concepts

Equilateral Triangle Properties

An equilateral triangle is a special type of triangle where all three sides are of equal length. Because of this unique property, all the internal angles are also equal, each measuring 60 degrees. This symmetry gives equilateral triangles some interesting properties:

• All sides are congruent, which means the triangle is perfectly symmetrical around its center.
• All angles are equal, simplifying the calculations when dealing with trigonometric functions or geometric proofs.
• The altitude, median, and angle bisector from any vertex to the opposite side are the same line. This means in an equilateral triangle, these segments are identical and meet at the triangle's centroid, which also serves as the orthocenter and circumcenter.

These properties make equilateral triangles simple yet powerful tools in geometric problems, providing an ideal model for studying symmetrical shapes.

Circumradius Formula

The circumradius is the radius of the circumscribed circle that passes through all vertices of the triangle. For an equilateral triangle, the formula to find the circumradius (R) is particularly neat:

• The formula is given by: \( R = \frac{a}{\sqrt{3}} \), where \( a \) is the length of the side of the triangle.
• This formula arises from the geometric properties of the equilateral triangle: since all sides and angles are uniform, the circumcenter (the center of the circumcircle) is positioned such that each side of the triangle can be easily connected by the radius of the circumcircle.
• In our specific problem, substituting \( a = 2\sqrt{3}\mathrm{~cm} \) gives \( R = \frac{2\sqrt{3}}{\sqrt{3}} = 2 \mathrm{~cm} \).

Understanding this formula helps in swiftly determining the circumradius, a crucial step in many geometric solutions that involve equilateral triangles and their properties.

Geometry

Geometry is an essential branch of mathematics that deals with the shapes, sizes, and properties of figures and spaces. Understanding geometry involves grasping various formulas and principles that apply to different shapes, including triangles.

• In geometry, triangles are fundamental building blocks because their properties are widely applicable in both theoretical proofs and practical applications.
• Geometric concepts, like the circumradius, help to contextualize how shapes behave in space, guiding students through complex systems and simplifying them to basic principles.
• The study of geometry involves not only memorizing formulas such as the circumradius formula for equilateral triangles but also developing a deep understanding of why these formulas work and how they relate to other mathematical concepts.

By exploring geometric problems, students sharpen their critical thinking skills and are better prepared for advanced mathematical study.