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Coordinate geometry, also known as analytic geometry, is a significant branch of mathematics. It uses a coordinate system to describe geometric figures. The basic idea is to translate geometric problems into algebraic equations using points defined by coordinates on a plane.

Each point in this plane is represented by a pair of numerical values - the coordinates, typically denoted as \((x, y)\).

• The x-coordinate determines the distance along the horizontal axis.
• The y-coordinate specifies the distance along the vertical axis.

This coordinate system allows us to solve various geometry problems, such as finding distances, slopes, and midpoints, by leveraging algebraic methods. Understanding coordinate geometry lays the groundwork for studying more complex mathematical concepts, as well as practical applications in numerous fields like computer graphics and robotics.

The midpoint of a line segment in coordinate geometry is an essential concept. A line segment has a definite beginning and endpoint and its midpoint is a point that divides the segment into two equal parts.

To find the midpoint between two points, say \(A(x_1, y_1)\) and \(B(x_2, y_2)\), we employ the midpoint formula:\[M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This formula calculates the average of the x-coordinates and the y-coordinates of the two endpoints.

• The x-coordinate of the midpoint is \(\frac{x_1 + x_2}{2}\), the average of the two x-coordinates.
• The y-coordinate of the midpoint is \(\frac{y_1 + y_2}{2}\), the average of the two y-coordinates.

The midpoint effectively represents the "center" of the line segment, providing a clear point of symmetry between the two endpoints.

To calculate the midpoint of a line segment connecting points \(A(-1,0)\) and \(B(-8,5)\), follow these simple steps.

**Step 1: Understand the Formula** The midpoint formula is \[M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]. Memorizing this basic formula is crucial for quick calculations.**Step 2: Insert the Coordinates** Identify the coordinates of the given points:

• For point \(A\), \(x_1 = -1\) and \(y_1 = 0\).
• For point \(B\), \(x_2 = -8\) and \(y_2 = 5\).

Plug these values into the formula:\[M \left( \frac{-1 + (-8)}{2}, \frac{0 + 5}{2} \right)\].**Step 3: Perform the Calculations** Calculate separately for the x and y coordinates:

• Compute the x-coordinate: \(\frac{-1 + (-8)}{2} = \frac{-9}{2} = -4.5\).
• Compute the y-coordinate: \(\frac{0 + 5}{2} = \frac{5}{2} = 2.5\).

Thus, the midpoint \(M\) of the segment is \(M(-4.5, 2.5)\), effectively dividing the line segment between points \(A\) and \(B\) into two equal parts.