Problem 12 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Let \(A_{0} A_{1} A_{2} A_{3} A_{4} A_{5}\) be a regular hexagon inscribed in a circle of unit radius. The product of length of the line segment \(A_{0} A_{1}, A_{0} A_{2}, A_{0} A_{4}\) is (a) \(\frac{3}{4}\) (b) \(3 \sqrt{3}\) (c) 3 (d) \(\frac{3 \sqrt{3}}{2}\)

Step-by-Step Solution

Answer

The product of the lengths of the line segments \(A_{0} A_{1}\), \(A_{0} A_{2}\), \(A_{0} A_{4}\) is \(2\sqrt{3}\), so none of the options are correct.

Step 1: Determine the side length of the hexagon

The side length of a regular hexagon with a circumscribed circle of radius 1 is also 1. Therefore, length of \(A_{0} A_{1}\) is 1.

Step 2: Determine the length of \(A_{0} A_{2}\)

The line segment \(A_{0} A_{2}\) forms a triangle with \(A_{0} A_{1}\) and \(A_{1} A_{2}\), which is an equilateral triangle as the hexagon is regular. Thus, the length of \(A_{0} A_{2}\) is \(2*cos(30^{\circ}) = \sqrt{3}\)

Step 3: Determine the length of \(A_{0} A_{4}\)

The line segment \(A_{0} A_{4}\) is a diameter of the circumscribed circle, thus its length is twice the radius which is 2.

Step 4: Calculate the product of lengths

The product of the lengths of line segments \(A_{0} A_{1}\), \(A_{0} A_{2}\), \(A_{0} A_{4}\) is \(1 * \sqrt{3} * 2 = 2\sqrt{3}\)

Key Concepts

Regular Hexagon Properties

Understanding the properties of a regular hexagon is crucial in many geometry problems. A regular hexagon is a six-sided polygon with all sides and angles being equal. In a regular hexagon inscribed in a circle, the radius of the circle is equal to the length of each side of the hexagon.

This relationship makes calculations within the hexagon straightforward, given that the internal angles are each 120 degrees. Since a hexagon can be divided into six equilateral triangles, each central angle, or the angle subtended at the center by each side, measures 60 degrees. When such a hexagon is inscribed in a circle, these angles are crucial for solving problems involving trigonometric ratios.

Circumscribed Circle

Moving on to the concept of a circumscribed circle, or what is often referred to in geometry as the 'circumcircle'. It is a circle that passes through all the vertices of a polygon, which in this case, is a hexagon. The center of this circle is also the center of the polygon, and the radius is referred to as the circumradius.

For a regular hexagon, the radius of the circumcircle is the same as the side length of the hexagon. In problems involving trigonometry, the circumradius provides a basic yet important relationship when dealing with the sine, cosine, and tangent functions, particularly since the radius is often used as the hypotenuse of a right-angled triangle within the hexagon.

Trigonometric Ratios

Trigonometric ratios are essential tools for solving geometry problems, especially those involving circles and regular polygons. These ratios - sine, cosine, and tangent - are based on the angles and sides of right-angled triangles. In the context of a regular hexagon inscribed in a circle, these ratios help us determine the lengths of sides and the relationship between different segments of the hexagon.

For example, in the given problem, the length of the segment connecting two non-adjacent vertices was found using the cosine of 30 degrees because this angle is associated with an equilateral triangle formed within the hexagon. Similarly, recognizing that the segment connecting a vertex to the opposite one is a diameter of the circle, twice the radius, is crucial. This is where a basic understanding of trigonometric ratios complements the geometric properties of the regular hexagon and its circumscribed circle.