Problem 91 - Algebra A Combined Function

Question

Find the midpoint of each line segment whose endpoints are given. (6,3)\(;(-1,-3)\)

Step-by-Step Solution

Answer

The midpoint of the line segment is (2.5, 0).

Step 1: Understanding the Midpoint Formula

The midpoint of a line segment is the point that divides the segment into two equal parts. The coordinates of the midpoint, given endpoints \((x_1, y_1)\) and \((x_2, y_2)\), can be found using the formula: \, \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

Step 2: Identifying Endpoints

For the given endpoints \((6,3)\) and \((-1,-3)\), identify \(x_1 = 6\), \(y_1 = 3\), \(x_2 = -1\), and \(y_2 = -3\).

Step 3: Calculating the Midpoint's X-coordinate

Use the midpoint formula to find the x-coordinate of the midpoint: \[ \frac{x_1 + x_2}{2} = \frac{6 + (-1)}{2} = \frac{6 - 1}{2} = \frac{5}{2} = 2.5 \]

Step 4: Calculating the Midpoint's Y-coordinate

Similarly, use the midpoint formula to find the y-coordinate of the midpoint:\[ \frac{y_1 + y_2}{2} = \frac{3 + (-3)}{2} = \frac{3 - 3}{2} = \frac{0}{2} = 0 \]

Step 5: Writing the Midpoint as a Coordinate Pair

Combine the x-coordinate and y-coordinate to express the midpoint of the line segment as a coordinate pair: \((2.5, 0)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In this system, we use ordered pairs of numbers, called coordinates, to define points on a plane. This approach allows us to calculate various geometric properties and relationships using algebraic methods.

Coordinate geometry is foundational in understanding geometric relationships and shapes; it bridges the gap between algebra and geometry. One of the simplest yet powerful applications of coordinate geometry is finding midpoints, distances, and slopes using coordinates. In our case, the midpoint of a line segment is calculated using the coordinates of its endpoints.

Line Segment

In coordinate geometry, a line segment is a part of a line that has two distinct endpoints. Unlike a line that extends indefinitely in both directions, a line segment is finite and measurable. The endpoints are crucial because they determine the segment's length and position in the coordinate plane.

To find important properties like the midpoint, we need these endpoints' coordinates. When we calculate the midpoint, we essentially find the point that is equidistant from both endpoints, providing a perfect balance along the segment.

X-Coordinate

The x-coordinate, also known as the abscissa, is the first number in an ordered pair and indicates a point's horizontal position on the coordinate plane. To emphasize the formula:

• For the midpoint x-coordinate, take the sum of the x-coordinates of the endpoints and divide by 2.
• In our example: \( \frac{x_1 + x_2}{2} = \frac{6 + (-1)}{2} = 2.5 \)

This result tells us how far the midpoint is from the origin along the horizontal axis. The calculation is straightforward and is generally a precursor to finding the complete midpoint.

Y-Coordinate

The y-coordinate, known as the ordinate, is the second number in an ordered pair and points to a location's vertical position on a plane. Like the x-coordinate, finding it requires using a simple operation:

• Add the y-coordinates of the endpoints and divide by 2 to find the midpoint's y-coordinate.
• For the example provided: \( \frac{y_1 + y_2}{2} = \frac{3 + (-3)}{2} = 0 \)

This simply means that the midpoint lies exactly on the x-axis, with no vertical displacement from the origin. By calculating both the x-coordinate and y-coordinate, we could write the coordinate pair for the midpoint as (2.5, 0), fully defining the midpoint's precise location.