Problem 6 - 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

xy = 4

Step 1: Understand the Hexagon

In a regular hexagon, all sides and angles are equal. The sides are denoted as AB, BC, CD, DE, EF, and FA. The diagonals such as AD and CF are longer than the sides.

Step 2: Analyze AD in terms of sides

The diagonal AD spans across three sides of the hexagon. Since the hexagon is regular, AD can be expressed in terms of the side length, say that AD = 2 times the side length of the hexagon, AB.

Step 3: Analyze CF in terms of sides

Similarly, the diagonal CF also spans across three sides of the hexagon. Hence, CF can also be expressed as 2 times the side length of the hexagon, AB.

Step 4: Use Given Ratios

Given that AD = x BC and CF = y AB, we already established that AD = 2AB and CF = 2AB. Then substituting, we get x BC = 2 AB and y AB = 2 AB.

Step 5: Solve for x and y

Since the hexagon is regular and BC = AB, substitute into x BC = 2 AB, we get x AB = 2 AB, which simplifies to x = 2. For the equation y AB = 2 AB, we have y = 2.

Step 6: Calculate x * y

Finally, substitute the values of x and y we found into the expression xy. So, xy = 2 * 2.

Key Concepts

Diagonals in Hexagons

In a regular hexagon, like the hexagon labeled as \(ABCDEF\), each vertex connects to five other vertices. This creates multiple diagonals within the hexagon itself. A diagonal in a hexagon is essentially a line segment that connects two non-adjacent vertices.

The key diagonals in a regular hexagon traverse across other sides. For example, when looking at diagonal \(AD\), it bypasses two vertices, connecting vertex \(A\) with vertex \(D\). This makes it two side lengths longer than adjacent sides. Similarly, diagonal \(CF\) follows the same pattern.

In mathematical terms for a regular hexagon:

• The diagonals span over three sides;
• Thus, each major diagonal, like \(AD\) or \(CF\), is twice the length of a single side.

Recognizing the role of diagonals is vital when solving problems that involve diagonal length ratios.

Hexagon Side Calculation

Calculating the properties of a regular hexagon involves understanding that all sides are equal in length. The challenge often lies in relating these side lengths to diagonals like \(AD\) and \(CF\).

In the problem, both \(AD\) and \(CF\) are described in terms of a single side of the hexagon, \(AB\). By analyzing their role, we see that

• \(AD = 2 \times \) the side length of the hexagon;
• Similarly, \(CF = 2 \times \) the side length of the hexagon.

These relationships prove useful when further calculations involve ratios like \(x\) and \(y\). Side calculations often unlock the answers to geometric ratios and are pivotal in verifying consistency across all sides and diagonals of the hexagon.

Equilateral Angles in Hexagon

A crucial component in understanding the geometry of a regular hexagon is equilateral angles. Each interior angle in a regular hexagon is equal, creating symmetrical and harmonious geometric properties.

Consider the interior angles:

• Each angle is \(120^\circ\);
• The angles between diagonals and sides reflect this symmetry.

Such congruent angles ensure that diagonal calculations and side computations are based on consistent patterns and equal lengths. The equilateral properties simplify complex-looking problems, providing straightforward pathways to solutions. Whether analyzing sides, angles, or diagonals, recognizing this symmetry helps in calculating unknown quantities effectively.