Problem 25 - 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. $$(0,7)\text { and }(4,-9)$$

Step-by-Step Solution

Answer

The midpoint is at (2, -1).

Step 1: Understanding the Midpoint Formula

To find the midpoint of a line segment that connects two points, \(x_1, y_1\) and \(x_2, y_2\), we use the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] where \(M\) represents the midpoint coordinates.

Step 2: Substitute the Given Points into the Formula

The given points are \( (0, 7) \) and \( (4, -9) \). Plug these values into the midpoint formula: \[ M = \left( \frac{0 + 4}{2}, \frac{7 + (-9)}{2} \right) \] which simplifies to \[ M = \left( \frac{4}{2}, \frac{-2}{2} \right) \]

Step 3: Calculate the Midpoint Coordinates

Now, calculate the values: \[ M_x = \frac{4}{2} = 2 \] \[ M_y = \frac{-2}{2} = -1 \] Therefore, the coordinates of the midpoint are \( (2, -1) \).

Key Concepts

Calculating Midpoint

The midpoint of a line segment is located exactly halfway between the two endpoints. It provides an average location of the endpoints. The formula for calculating the midpoint is quite simple and helpful in coordinate geometry. It is defined as:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]where:

• \(M\) denotes the midpoint coordinates.
• \(x_1, y_1\) are the coordinates of the first point.
• \(x_2, y_2\) are the coordinates of the second point.

For example, if you need to find the midpoint of the segment joining points \((0, 7)\) and \((4, -9)\), you simply substitute these values into the formula:\[ M = \left( \frac{0+4}{2}, \frac{7 + (-9)}{2} \right) = (2, -1) \]This means point \((2, -1)\) is centrally located between the two given points.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry through the use of coordinates on a plane. Every point on this plane is given by a pair of numbers, which enables you to analyze geometrical shapes using algebraic equations. This blending of subjects allows us to solve complex problems about lines, curves, and geometric shapes in a tangible manner.Key aspects of coordinate geometry:

• Coordinates: Every point is represented by an ordered pair \((x, y)\).
• Line Segments: The shortest distance between two points, definable using endpoints.
• Distance and Midpoints: Use mathematical formulas to determine distances and midlocations on lines.

In exercises like finding the midpoint, understanding coordinate geometry is essential as it provides the foundation and techniques to calculate accurately on the plane.

Line Segments

A line segment is part of a line bounded by two distinct endpoints, clearly defined in mathematics to differentiate between infinite lines and finite distances. In the context of coordinate geometry, a segment is crucial because it can be calculated and measured, unlike a line that extends indefinitely.Key characteristics of a line segment:

• Endpoints: Points like \((0, 7)\) and \((4, -9)\) that define the beginning and end of a segment.
• Length: Measured by the distance formula, signifying the distance between the two endpoints.
• Midpoint: An indicator of the halfway mark, providing balance along the line segment.

When you are calculating the midpoint of a line segment using the midpoint formula, you effectively find the average of the endpoints, offering insight into the segment's position on the plane.