Problem 5 - A Complete Resource Book in Mathematics for JEE Main 2017

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

C

Step 1: Understand a Regular Hexagon

A regular hexagon has equal side lengths and equal internal angles. Thus, for hexagon \(ABCDEF\), each side, \(AB, BC, CD, DE, EF, FA\), is equal.

Step 2: Relationships in the Hexagon

Given that \(AD = xBC\) and \(CF = yAB\). We need to find \(x\) and \(y\) such that their product \(xy\) satisfies the equation. Notice \(AD\) and \(CF\) are not sides of the hexagon but diagonals.

Step 3: Diagonal Lengths in Hexagon

For a regular hexagon of side length \(s\), the length of the longest diagonal, which skips two vertices, such as \(AD\) is twice the side length, \(2s\). The shortest diagonal, such as \(CF\), that skips one vertex, is equal to the side length.

Step 4: Substitute Values into Given Equations

Using the above information, substitute the lengths into the given equations:\[AD = 2s = xBC = xs\]\[CF = s = yAB = ys\]Therefore, we have \(x = 2\) and \(y = 1\).

Step 5: Calculate the Product xy

Now calculate the product \(xy = 2 \cdot 1 = 2\).

Step 6: Verify the Solution

Both \(AD\) and \(CF\) are confirmed to be diagonals of the regular hexagon with the correct proportional values, ensuring \(xy = 2\).

Key Concepts

Regular Hexagon

A regular hexagon is a six-sided polygon where all sides and all internal angles are equal. This means each interior angle is 120 degrees, contributing to the symmetrical and balanced nature of the shape. Regular hexagons appear frequently in nature and design, such as in honeycombs.
In mathematics, these hexagons offer a wealth of properties. For example:

• The perimeter can be calculated by multiplying the side length by 6.
• The area can be found using the formula: \[ \text{Area} = \frac{3\sqrt{3}}{2} s^2 \] where \( s \) is the side length.

Understanding the properties of a regular hexagon is fundamental in solving geometric problems related to it.

Diagonal Lengths

In a regular hexagon, different lengths of diagonals exist. Diagonals are lines connecting non-adjacent vertices. For a hexagon with side length \( s \), we have two primary types of diagonals:

• Longest Diagonal: This skips two vertices, like from point \( A \) to point \( D \) (\( AD \)). This diagonal is calculated as \( 2s \).
• Shortest Diagonal: These skip one vertex, like \( CF \), and are equal to the side length \( s \).

These diagonal expressions help in solving spatial problems within hexagons. By understanding these types of diagonals, students can navigate through geometric exercises with complex internal configurations.

Side Length

The side length is crucial in defining all aspects of a regular hexagon, as it serves as a unit measure for all other quantities.

• Every side in a regular hexagon is equal, creating a high degree of symmetry and uniformity throughout the shape.
• Many geometric properties depend directly on the side length: the perimeter is \( 6s \), while key diagonals can be expressed in terms of \( s \) (e.g., the longest diagonal \( 2s \) and the shortest \( s \)).
• When problems specify diagonal relationships, as seen in the exercise, the side length serves as a fundamental unit for converting dimensions and validating relationships like \( x = 2 \) and \( y = 1 \).

By thoroughly understanding how side length determines other properties, students bolster their ability to manage complex geometric scenarios effectively.