Problem 9 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A square, of each side 2 , lies above the \(x\)-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle \(30^{\circ}\) with the positive direction of the \(x\)-axis, then the sum of the \(x\)-coordinates of the vertices of the square is : \(\quad\) [Online April 9, 2017] (b) \(2 \sqrt{3}-2\) (c) \(\sqrt{3}-2\) (d) \(\sqrt{3}-1\) (a) \(2 \sqrt{3}-1\)

Step-by-Step Solution

Answer

The sum of the x-coordinates of the vertices is \(2\sqrt{3} - 2\).

Step 1: Identify the coordinates of the rotated vertex

Given that one vertex is at the origin (0,0), and the square is rotated by an angle of \(30^{\circ}\) from the x-axis, the other vertex on this line can be found using trigonometric functions. Since each side of the square is 2, the coordinates of this point are \((2\cos 30^{\circ}, 2\sin 30^{\circ})\). Calculating: \(2\cos 30^{\circ} = 2\times \frac{\sqrt{3}}{2} = \sqrt{3}\) and \(2\sin 30^{\circ} = 2\times \frac{1}{2} = 1\). Thus, the coordinates are \((\sqrt{3}, 1)\).

Step 2: Determine the coordinates of other vertices

Assuming the square is oriented such that both sides are lying at \(30^{\circ}\) and \(120^{\circ}\) with the x-axis. For the side at \(120^{\circ}\), the vertex coordinate can be calculated as \((2\cos 120^{\circ}, 2\sin 120^{\circ})\). Calculating these gives \((-2 \cos 60^{\circ}, 2 \sin 60^{\circ}) = (-1, \sqrt{3})\).

Step 3: Calculate the last vertex of the square

For the last corner, add the coordinates of a parallel line orthogonally offset from the \((\sqrt{3}, 1)\) vertex. As perpendicular to \(30^{\circ}\) is \(120^{\circ}\), the offset point is \((\sqrt{3} - 1, 1 + \sqrt{3})\).

Step 4: Add the x-coordinates of all vertices

Sum the x-coordinates of all vertices: The origin (0), \((\sqrt{3}, 1)\) provides \(\sqrt{3}\), \((-1, \sqrt{3})\) provides \(-1\), and \((\sqrt{3} - 1, 1 + \sqrt{3})\) provides \((\sqrt{3}-1)\). The sum is \(0 + \sqrt{3} - 1 + \sqrt{3} - 1 = 2\sqrt{3} - 2\).

Step 5: Select the correct answer

From the given options, the correct answer is \(b)\; 2\sqrt{3}-2\), which matches our calculated sum of x-coordinates.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In this problem, we are dealing with a square positioned in the coordinate plane, where one vertex starts at the origin

• The origin is defined as the point (0,0) where the x-axis and y-axis intersect.
• The coordinates help in expressing geometric shapes algebraically, allowing calculations like rotation and translation.

In this exercise, understanding the rotation of a square and the coordinates resulting from that alteration is key. The square is rotated 30 degrees from the x-axis. This means you can use trigonometric functions to find the other vertices, as these rely on angles measured with respect to the x-axis. Using sine and cosine for such rotations simplifies the calculation of new coordinates. This captures the essence of geometric transformation in a coordinate system.

Trigonometry

Trigonometry deals with the angles and sides of triangles, but it is extremely useful when dealing with rotation and movement on the coordinate plane. In this case, trigonometry helps define the positions of a square's vertices when one side is rotated by 30 degrees.

• We use the trigonometric functions cosine (\( \cos \)) and sine (\( \sin \)) for transformation.
• The cosine of the angle provides the x-coordinate shift from a point, while the sine gives the y-coordinate shift.

For example, if we have a side length of 2 units and an angle of 30 degrees, the new coordinate from the original is found by multiplying the side length by the cosine of 30 degrees for the x direction and the sine of 30 degrees for the y direction.

• These calculated points are key to understanding the position of geometrically transformed shapes.

Trigonometry thus serves as a bridge between algebraic calculations and spatial geometry, guiding us through rotating figures on the coordinate plane.

Square Properties

A square is a special type of rectangle with all four sides of equal length. Understanding the properties of squares helps in solving problems involving rotation, symmetry, and coordinates.

• Each angle in a square is 90 degrees.
• All diagonals in a square are equal in length, making it a highly symmetrical figure.

In this exercise, the square has a side of 2 units. When solving for rotated shapes, it's crucial to keep in mind that despite the orientation, the lengths remain the same. What changes is the direction of the sides, governed by the angles with the x-axis. Recognizing that a square's properties do not change ensures consistent problem-solving.

• The movement and rotation of squares require careful calculation, balancing geometric and algebraic methods.