Problem 60 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If in a regular polygon the number of diagonals is 54 , then the number of sides of this polygon is (a) 12 (b) 6 (c) 10 (d) 9

Step-by-Step Solution

Answer

The number of sides is 12.

Step 1: Understand the formula for diagonals

To find the number of diagonals in a regular polygon with \( n \) sides, we use the formula: \( \text{Number of diagonals} = \frac{n(n-3)}{2} \).

Step 2: Set up the equation for given diagonals

We know the polygon has 54 diagonals. Therefore, the equation becomes \( \frac{n(n-3)}{2} = 54 \).

Step 3: Simplify and solve for n

Multiply both sides by 2 to clear the fraction: \( n(n-3) = 108 \). Expand and rearrange to form a quadratic equation: \( n^2 - 3n - 108 = 0 \).

Step 4: Solve the quadratic equation

Apply the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -3 \), and \( c = -108 \). Substitute these values to get \( n = \frac{3 \pm \sqrt{9 + 432}}{2} \). Simplify under the square root: \( n = \frac{3 \pm \sqrt{441}}{2} \), which gives \( n = \frac{3 \pm 21}{2} \).

Step 5: Evaluate the possible solutions

The possible solutions are \( n = \frac{24}{2} = 12 \) and \( n = \frac{-18}{2} = -9 \). However, \( n \) must be a positive integer, so \( n = 12 \).

Step 6: Verify the solution

Substitute \( n = 12 \) back into the diagonal formula: \( \frac{12(12-3)}{2} = \frac{12 \times 9}{2} = 54 \). Since this matches the given number of diagonals, the solution is correct.

Key Concepts

Regular Polygon

A regular polygon is a fascinating geometrical shape where all angles are equal and all sides have the same length. Imagine a perfect pentagon or an equilateral triangle—those are regular polygons. These shapes are often studied because they are symmetrical and easy to analyze mathematically.
Understanding how regular polygons behave can be crucial in fields like architecture and design where symmetry is often key.
Some common characteristics of regular polygons include:

• Equidistant vertices: All the vertices (corners) are evenly spaced around a central point.
• Equal angles: Every internal angle is the same, which often simplifies calculations.
• Circumscribed and inscribed circles: Regular polygons can fit perfectly inside a circle (inscribed) or have a circle fit perfectly around them (circumscribed).

Knowing these properties helps in solving for various unknowns, such as the number of sides when the number of diagonals is known.

Quadratic Equation

Quadratic equations are a cornerstone of algebra and are essential in solving many geometrical problems, like finding missing sides in a regular polygon. These equations take the form \[ ax^2 + bx + c = 0 \]. They often appear when dealing with problems that require finding unknown values that are squared or that follow a curve.
You can find solutions to a quadratic equation using several methods:

• Factoring: Splitting the equation into simpler products that can easily be solved.
• Completing the square: Re-writing the quadratic in the form \( (x + p)^2 = q \)
.
• Quadratic formula: This is a reliable way to find the solution: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

Each method has its advantages, but the quadratic formula is often used since it works for any quadratic, providing quick and reliable solutions.

Diagonal Formula

The diagonal formula is a crucial tool in polygon geometry, helping to determine how many diagonals a polygon has. In a polygon, a diagonal is a line segment that connects two non-adjacent vertices. For a regular polygon with \( n \) sides, the number of diagonals can be found using the formula:\[ \text{Number of diagonals} = \frac{n(n-3)}{2} \]This formula is derived by considering that each vertex can connect to \( n-3 \) other vertices (since it can't connect to itself or the two adjacent vertices) and dividing by two to avoid double-counting each diagonal.
Applying this formula can also help ascertain other properties of the polygon, such as validating whether a given number of diagonals corresponds correctly to a certain number of sides, as we did in the exercise when determining \( n = 12 \) for 54 diagonals. Understanding this formula and its derivation allows for deeper insights into polygon geometry and spatial reasoning.