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Coordinate Geometry is a branch of geometry where we use coordinate planes to analyze geometric locations. It involves placing points on the plane using ordered pairs, known as coordinates. Each point on a coordinate plane is represented by a pair

• The first number in the pair denotes the horizontal position (x-coordinate).
• The second number gives the vertical position (y-coordinate).

In our exercise, points E and F are given as - E(-3, -2) - F(5, 8) These coordinates help us easily find relationships between points, such as the distance between them or the midpoint of a line segment that connects them, using simple algebraic formulas.

The Distance Formula is a tool in coordinate geometry used to find the distance between two points in a coordinate plane. We use it when points are defined by their coordinates To use the distance formula, follow these steps:

• Identify the coordinates of the points, say - Point 1 as \((x_1, y_1)\)
• Point 2 as \((x_2, y_2)\)

The distance \(d\)\(\) can be calculated using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In our example, substituting the coordinates E(-3, -2) and F(5, 8) into the formula gives\[d = \sqrt{(5 - (-3))^2 + (8 - (-2))^2}\]By simplifying further, we find the distance is \(\)\(\)\(\)\(\)\(\)\(\)\[\sqrt{164} \approx 12.81\]This formula effectively uses the Pythagorean theorem to establish the shortest distance between two points.

The Midpoint Formula allows us to determine the point that divides a line segment into two equal parts. It's particularly useful in coordinate geometry for identifying the center point between two endpoints.When you want to find a midpoint, follow these simple steps:

• Start with the coordinates of points A and B defined as:
• Point A \((x_1, y_1)\)
• Point B \((x_2, y_2)\)

The midpoint \(M\)\(\) can be calculated using:\[M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\]For example, for points E(-3, -2) and F(5, 8), the calculation looks like this:\[M = \left(\frac{-3+5}{2}, \frac{-2+8}{2}\right) = (1, 3)\]So, the coordinates of the midpoint are \((1, 3)\). This formula simplifies the process of averaging the coordinates, providing a quick way to find the balanced point on the line segment.