Four points \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right),\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right),\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right)\) and \(\left(\mathrm{x}_{4}, \mathrm{y}_{4}\right)\) are such that \({ }^{4} \sum_{\mathrm{i}=1}\left(\mathrm{x}_{\mathrm{i}}^{2}+\mathrm{y}_{\mathrm{i}}^{2}\right) \leq 2\left(\mathrm{x}_{1} \mathrm{x}_{3}+\mathrm{x}_{2} \mathrm{x}_{4}+\mathrm{y}_{1} \mathrm{y}_{2}+\mathrm{y}_{3} \mathrm{y}_{4}\right)\). Then these points are vertices of (a) Parallelogram (b) Rectangle (c) Square (d) Rhombus

The concept of a parallelogram is fundamental in understanding various types of quadrilaterals. In Euclidean geometry, a parallelogram is defined as a four-sided shape where opposite sides are parallel and equal in length, and opposite angles are equal. The parallelogram also has interesting properties such as its diagonals bisecting each other and each diagonal dividing the parallelogram into two congruent triangles.

To visualize this, imagine a four-sided figure where any one side can be translated along a line parallel to itself to overlap with the opposite side. Two pairs of sides with this characteristic automatically make the figure a parallelogram. The exercise floats the possibility that the points given could form a parallelogram, which can be discerned by the properties mentioned and applying the distance formula to calculate side lengths and diagonal lengths for verification. Identifying these properties helps in solving problems related to quadrilaterals in competitive exams like the JEE, where a methodical approach is crucial.

Geometry plays a significant role in the Joint Entrance Examination (JEE), one of India's highly competitive engineering entrance tests. Topics such as quadrilaterals and their properties become essential learning materials since they have practical applicability in solving complex problems often found in the JEE. In geometry-related problems, it's not just about knowing the concepts but also about applying them efficiently.

For instance, knowing the defining properties of quadrilaterals like squares, rectangles, parallelograms, and rhombuses is key. JEE aspirants are expected to manipulate these shapes, often represented as coordinates on a plane, to find their areas, perimeters, angles, and more. As we saw in the given exercise, the application of the distance formula, along with algebraic manipulation, is a powerful tool to identify which type of quadrilateral is formed.

The distance formula is a derivative of the Pythagorean theorem that calculates the distance between two points in a coordinate system. It is given by the equation:
\[d(A,B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
In essence, if you have two points A and B with coordinates \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the distance formula lets you compute the 'straight line' distance between them. This formula is extremely useful for determining lengths of sides and diagonals of quadrilaterals when only coordinates are known.

Moreover, the distance formula shines in its simplicity and wide applicability in problems involving the geometry of shapes. To truly grasp this concept, imagine plotting points on a graph and drawing a right triangle whose hypotenuse is the line segment between the points; the distance formula essentially finds the length of this hypotenuse. It's an indispensable tool for geometric proofs and for confirming the properties of shapes such as parallelograms within coordinate geometry-related questions, such as those tackled by students preparing for exams like the JEE.