Find the Midpoint of a Line Segment using the Midpoint Formula

This problem asks us to find the coordinates of the midpoint of the line segment connecting two specific points: \((-4, 7)\) and \((2, 3)\). Finding the midpoint is a fundamental concept in coordinate geometry.

Understanding the Midpoint Concept

The midpoint of a line segment is the point exactly halfway between the two endpoints. It divides the segment into two equal parts. In a coordinate plane, we can find the coordinates of this midpoint using a specific formula.

Applying the Midpoint Formula

The midpoint formula is used to find the coordinates \((x_m, y_m)\) of the midpoint of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\). The formula is:

\(\left(x_m, y_m\right) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)

This means we average the x-coordinates and average the y-coordinates of the two endpoints separately to find the coordinates of the midpoint.

Step-by-Step Midpoint Calculation

Let the given points be \(P_1 = (-4, 7)\) and \(P_2 = (2, 3)\).

We can assign the coordinates:

• \(x_1 = -4\)
• \(y_1 = 7\)
• \(x_2 = 2\)
• \(y_2 = 3\)

Now, we apply the midpoint formula:

1. Calculate the x-coordinate of the midpoint (\(x_m\)):

   \(x_m = \frac{x_1 + x_2}{2} = \frac{-4 + 2}{2}\)

   \(x_m = \frac{-2}{2} = -1\)

2. Calculate the y-coordinate of the midpoint (\(y_m\)):

   \(y_m = \frac{y_1 + y_2}{2} = \frac{7 + 3}{2}\)

   \(y_m = \frac{10}{2} = 5\)

So, the coordinates of the midpoint of the segment joining \((-4, 7)\) and \((2, 3)\) are \((-1, 5)\).

Comparing Calculated Midpoint with Options

Let's compare our calculated midpoint coordinates with the given options.

The calculated midpoint \((-1, 5)\) matches the coordinates provided in Option 1.

Revision Table: Essential Coordinate Geometry Formulas

Here is a quick recap of some essential formulas used in coordinate geometry when dealing with points and segments:

Additional Information: Introduction to Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is a branch of mathematics that uses coordinates to relate algebraic equations and geometric figures. It allows us to solve geometrical problems using algebraic methods.

Key ideas in coordinate geometry include:

• Representing points using ordered pairs \((x, y)\) on a coordinate plane.
• Defining geometric shapes (like lines, circles, parabolas) using algebraic equations.
• Using formulas to calculate properties of figures, such as the distance between points, the midpoint of a segment, the slope of a line, and the area of a polygon.

Understanding the midpoint formula is a crucial step in mastering coordinate geometry problems involving line segments. It's frequently used in various geometry and algebra applications.