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The midpoint formula is a fundamental concept in coordinate geometry used to find the exact middle point between two given points on a coordinate plane. It's like finding a balance point between two locations. The formula itself is very straightforward and is given by:

• For any two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint \( M \) is calculated as:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This formula computes the mean or average of the x-coordinates and the y-coordinates separately to find their center. By doing this, it gives the midpoint coordinates that are equidistant from each input point.
Understanding and applying this formula is crucial when dealing with problems related to dividing lines into equal parts.

The coordinate plane, also known as the Cartesian plane, is a two-dimensional space where each point is specified by a pair of numerical coordinates. These coordinates are usually written as \( (x, y) \), where \( x \) represents the horizontal distance from the origin, and \( y \) is the vertical distance.

• The plane is divided into four quadrants by the x-axis and y-axis, which intersect at the origin \( (0,0) \).
• In our example, Point A is located at \( (-1, 6) \) and Point B is at \( (-7, -2) \). The exercise involves determining a new point that lies midway between these given points.

The coordinate plane is essential for graphing and visualizing geometric problems, and being familiar with it helps in understanding how geometrical operations such as finding midpoints work.

To effectively calculate the midpoint, it is important to go through a series of simple steps. Let's recap the calculation process with clear steps:

Identify the Points

• Start by specifying the given points. In our scenario, these points are \( (-1, 6) \) and \( (-7, -2) \).

Use the Midpoint Formula

• Apply the formula \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \) to find the midpoint.

Calculate the Midpoint

• Substitute the x and y values from both points. For \( x \), calculate: \[ \frac{-1 + (-7)}{2} = -4 \] For \( y \), calculate: \[ \frac{6 + (-2)}{2} = 2 \]
• This results in the midpoint \( (-4, 2) \).

Following these structured steps ensures accuracy and reinforces the method, aiding in better understanding and recall for solving similar problems in the future.