Problem 5 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(A B C D E F\) be a regular hexagon. If \(A D=x B C\) and \(C F=y A B\), then \(x y=\) (A) 4 (B) \(-4\) (C) 2 (D) \(-2\)

Step-by-Step Solution

Answer

The product \(xy\) is 4.

Step 1: Understand the Problem Statement

To solve this problem, we need to use the geometric properties of a regular hexagon. We know that in a regular hexagon, all sides are equal and opposite sides are parallel. The problem provides two expressions involving the sides and diagonals of the hexagon and asks us to find the product \(xy\).

Step 2: Visualize and Label the Hexagon

Visualize the hexagon with points labeled as \(A, B, C, D, E, F\). In a regular hexagon, all side lengths are equal, say \(s\). The diagonals \(AD\) and \(CF\) need to be analyzed as they are crucial for solving the problem.

Step 3: Use Properties of a Regular Hexagon

In a regular hexagon, \(AD\) and \(CF\) are twice the length of the sides. This is because diagonal \(AD\) stretches from vertex \(A\) to vertex \(D\), skipping one vertex, making it double the side length.

Step 4: Express Given Equations in Terms of Side Length

From the problem, \(AD = x \cdot BC\) and \(CF = y \cdot AB\). Since \(AD = 2s\) and \(BC = AB = s\), these give equations, \(2s = x \cdot s\) and \(2s = y \cdot s\).

Step 5: Solve for x and y

Dividing both sides of the first equation by \(s\), we get \(x = 2\). Similarly, solving the second equation, we get \(y = 2\).

Step 6: Calculate xy

Now, calculate the product \(xy = 2 \cdot 2 = 4\). Thus, the value of \(xy\) is 4.

Key Concepts

Properties of a Hexagon

A hexagon is a six-sided polygon, and when it is regular, all its sides and angles are equal. This means that in a regular hexagon:

• All sides are of equal length.
• Each interior angle measures 120 degrees.
• Opposite sides are parallel to each other.

These properties allow us to easily determine lengths and relationships within the hexagon. For example, since each side is congruent, the symmetry of the hexagon simplifies calculations. The understanding of these properties helps in solving geometric problems, particularly those involving length calculations and transformations like rotations and reflections.

Geometric Properties

The geometric properties of a regular hexagon extend beyond just equal sides and angles. These properties include:

• Symmetry: A regular hexagon has six lines of symmetry and rotational symmetry of order six.
• Center of the Hexagon: The center can be used as a point of rotation or analysis for various geometric problems.
• Coordination with Circles: A regular hexagon can be circumscribed by a circle or inscribed in one. The circle encompassing a regular hexagon is termed as its circumcircle, and each vertex of the hexagon touches the circle.

These geometric properties make regular hexagons especially useful in solving complex geometric problems as they provide a simple point of reference or transformation.

Diagonals in Polygons

Diagonals are line segments that connect non-adjacent vertices of a polygon. In a regular hexagon, understanding its diagonals is key to solving geometrical problems. Here's why:

• A regular hexagon has nine diagonals.
• Diagonals can be categorized based on their lengths. For instance, in a regular hexagon, some diagonals stretch across the hexagon (like from vertex A to D), and are longer compared to others.
• In this exercise, the diagonals AD and CF each measure twice the side length, which helps in calculating ratios or transformations involving the hexagon's sides and diagonals.

Knowing how to identify and calculate the properties of these diagonals helps in solving exercises involving hexagons. By using the example of diagonals in a regular hexagon, we can better understand how geometry can be applied to solve polynomial problems efficiently.