Problem 23 - College Algebra

Question

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points. $$(-1,1)\text { and }(7,-4)$$

Step-by-Step Solution

Answer

The midpoint is (3, -1.5).

Step 1: Understanding the Midpoint Formula

The midpoint \((x_m, y_m)\) of a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \[ x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}. \] This formula calculates the average of the x-coordinates and the y-coordinates separately.

Step 2: Plug in the Given Values

For the points \((-1, 1)\) and \((7, -4)\), substitute \(x_1 = -1\), \(y_1 = 1\), \(x_2 = 7\), and \(y_2 = -4\) into the formulas: \[ x_m = \frac{-1 + 7}{2}, \quad y_m = \frac{1 + (-4)}{2}. \]

Step 3: Calculate the Midpoint’s x-coordinate

Proceed to calculate the x-coordinate:\[ x_m = \frac{-1 + 7}{2} = \frac{6}{2} = 3. \]

Step 4: Calculate the Midpoint’s y-coordinate

Now calculate the y-coordinate:\[ y_m = \frac{1 - 4}{2} = \frac{-3}{2} = -1.5. \]

Step 5: Conclusion

The coordinates of the midpoint are \((3, -1.5)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful mathematical tool that combines algebra and geometry. By using a coordinate plane, which consists of a horizontal x-axis and a vertical y-axis, we can represent geometric problems algebraically. This method makes it easier to analyze and solve problems involving shapes, lines, and points.

In the context of finding a midpoint, coordinate geometry helps us place points on this plane with exact coordinates, such as \((-1,1)\) and \((7,-4)\). From these coordinates, calculations like those in our example become manageable and precise.

This combination of algebra and geometry provides precise mathematical language and tools. This allows us to express complex geometric concepts through calculations. Every shape or line we draw can be analyzed or broken down with the help of coordinates, making it invaluable in modern mathematics.

Line Segment

A line segment is a part of a line defined by two endpoints. It differs from a line, which extends infinitely in both directions. In our example, these endpoints are \((-1,1)\) and \((7,-4)\).

Unlike lines, which continue indefinitely, line segments have fixed lengths. This makes them easier to handle in geometric calculations. Finding the midpoint of a line segment, as we've done in the exercise, is a typical problem in coordinate geometry.

The midpoint divides the line segment into two equal parts. Visually, if you imagine drawing a line on the coordinate plane from \((-1,1)\) to \((7,-4)\), the midpoint \((3,-1.5)\) can be seen as the center point of this segment. This central position helps us understand balance and symmetry in shapes and paths.

Averaging Coordinates

Averaging coordinates is the method we use to find the midpoint of a line segment. By averaging, we mean taking the mean of the x-coordinates and the y-coordinates separately.

To elaborate, given two points \((x_1, y_1)\) and \((x_2, y_2)\), their midpoint \((x_m, y_m)\) is calculated as follows:

• The x-coordinate of the midpoint, \(x_m\), is found by averaging the x-values: \(x_m = \frac{x_1 + x_2}{2}\).
• Similarly, the y-coordinate of the midpoint, \(y_m\), is determined by averaging the y-values: \(y_m = \frac{y_1 + y_2}{2}\).

The logic behind this is straightforward: averaging the x-values gives us a point that is centered between the two original x coordinates, and the same goes for the y-values.

This method ensures that both the x and y values of the midpoint accurately reflect the exact point that splits the segment equally. This simple averaging provides a versatile solution to locate midpoints easily and accurately on the coordinate plane.