Let \(\mathrm{A}(2,-3)\) and \(\mathrm{B}(-2,1)\) be vertices of a triangle \(\mathrm{ABC}\). If the centroid of this triangle moves on the line \(2 x+3 y=1\), then locus of the vertex \(\mathrm{C}\) is the line (a) \(2 \mathrm{x}+3 \mathrm{y}=9\) (b) \(2 x-3 y=7\) (c) \(3 \mathrm{x}+2 \mathrm{y}=5\) (d) \(3 x-2 y=3\)

The study of coordinate geometry involves placing geometric figures in a coordinate plane to measure and calculate their properties using algebra and the Cartesian coordinate system. It's a powerful tool that connects algebraic equations to geometric figures.

In the context of the centroid of a triangle, coordinate geometry allows us to use an algebraic approach to find a point that represents the average position of the triangle's vertices. By plotting the vertices of the triangle in the coordinate plane, we can use the formula for the centroid to calculate its exact location.

The coordinate plane is divided into four quadrants by a horizontal x-axis and a vertical y-axis. Each point in the plane is defined by an ordered pair of numbers \(x, y\), representing the point's horizontal and vertical positions, respectively.

The concept of 'locus' refers to the set of points that satisfy certain conditions or equations. In other words, it's the path a point follows as it moves according to a specific rule.

When given a geometric condition, such as the centroid of a triangle moving along a line, the locus tells us where the third vertex of the triangle can be placed so that the condition remains true. In the exercise, the centroid's locus was given, and from it, we deduced the possible locations of the third vertex C by using the properties of the centroid.

Understanding the locus of a point is critical in problems that deal with motion or constraints within geometric figures, as it allows us to visualize and calculate the positions of parts of figures in relation to the whole.

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. There are three medians in a triangle, and interestingly, they all intersect at a single point: the centroid.

The property that the centroid divides each median into two segments with a 2:1 ratio is significant in locating the centroid on a coordinate plane. This characteristic was used in the exercise to develop an equation for the coordinates of the centroid in terms of the triangle's vertices.

Understanding medians is crucial when working on problems related to triangle symmetry and balance. Since the centroid represents the center of mass or balance point, it serves as a key point in many applications, both theoretical and practical, such as in engineering and design.