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Coordinate geometry, also known as analytic geometry, is the study of geometric figures through the use of a coordinate system. This system allows for the precise positioning of points on a plane using pairs of numbers known as coordinates, typically represented as (x, y).

In coordinate geometry, each point on a plane is defined by its distance from two fixed perpendicular lines, known as the x-axis (horizontal) and the y-axis (vertical). The intersection of these axes is called the origin, designated as the point (0, 0).

The fundamental elements of coordinate geometry include points, line segments, and shapes constructed within this coordinate plane. By using algebraic formulas and principles, we can calculate distances, midpoints, slopes, and much more to understand the properties of various geometric figures.

A line segment is a part of a line bounded by two distinct end points, and the midpoint of a line segment is the point that divides the segment into two equal parts. In coordinate geometry, this midpoint can be thought of as the 'average' location between the two endpoints.

Finding the midpoint is crucial in many applications, including geometry, engineering, and computer graphics. The concept is visually akin to balancing a stick on your finger; the point where the stick balances evenly is the midpoint. It has the property that it is equidistant from both endpoints of the line segment.

The midpoint formula is a straightforward way to calculate the midpoint of a line segment when you know the coordinates of its endpoints. The formula is given by: \[Midpoint (M) = \left( \frac{x_{1} + x_{2}}{2} , \frac{y_{1} + y_{2}}{2} \right)\]This formula takes the average of the x-coordinates and the average of the y-coordinates of the endpoints to find the x and y coordinates of the midpoint, respectively.

To use the midpoint formula, follow these steps:

• Identify the coordinates of the endpoints.
• Substitute the coordinates into the formula.
• Simplify the expressions to find the coordinates of the midpoint.

For the given points (4, 7) and (-10, 7), the midpoint is found by averaging the x-coordinates (4 and -10) and the y-coordinates (7 and 7). As a result, the midpoint is (-3, 7).Understanding and applying the midpoint formula is essential in many fields that rely on geometric and spatial reasoning.