Question

The coordinates of the endpoints of a segment are given. Find the coordinates of the midpoint of each segment. \((6,0),(13,0)\)

Short answer

The midpoint of the segment is (9.5, 0).

Step-by-step solution

Understand the Midpoint Formula

The formula to find the midpoint \(M\) of a line segment given the endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).We will apply this formula to the given endpoints.

Substitute the Endpoint Coordinates

Given the endpoints \( (6, 0) \) and \( (13, 0) \), we need to substitute these coordinates into the midpoint formula.

Calculate the Midpoint's x-coordinate

Substitute \( x_1 = 6 \) and \( x_2 = 13 \) into the formula: \\( \frac{6 + 13}{2} = \frac{19}{2} = 9.5 \) \Thus, the x-coordinate of the midpoint is 9.5.

Calculate the Midpoint's y-coordinate

Substitute \( y_1 = 0 \) and \( y_2 = 0 \) into the formula: \\( \frac{0 + 0}{2} = 0 \) \Thus, the y-coordinate of the midpoint is 0.

Compile the Midpoint Coordinates

Using the calculated x and y coordinates, the midpoint of the segment is \( (9.5, 0) \).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to use algebraic techniques for solving geometric problems. It is a vital tool for finding various geometric properties, including the distance between points, the slopes of lines, and the midpoints of line segments.
Key Components of Coordinate Geometry

• Points: Defined in a two-dimensional plane with x and y coordinates, represented as \((x, y)\).
• Lines: Defined by a set of points in the coordinate plane. It can be described by equations like \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
• Distance and Midpoint: Using coordinate geometry, you can easily calculate distances between points and find the midpoint of a line segment by applying specific formulas.

The coordinate plane itself is divided into four quadrants by the x-axis and y-axis. In the given exercise, you're working in a simple one-dimensional scenario on the x-axis, which simplifies the calculations.

Line Segment

A line segment is a part of a line that has two endpoints. Unlike a line, which stretches infinitely in both directions, a line segment is fixed between two points.
Characteristics of Line Segments

• Endpoints: The two points where the segment begins and ends. For a segment on a graph, these endpoints are denoted by coordinates, such as \((6, 0)\) and \((13, 0)\).
• Length: The distance between the two endpoints, which can be calculated using the distance formula.
• Fixed Position: The segment does not extend beyond its endpoints, making it the simplest form of a geometric figure in a coordinate plane.

Understanding line segments is crucial when calculating midpoints because the midpoint is the point that divides the segment into two equal parts. In coordinate geometry, once you have the endpoints, you can easily proceed to find the midpoint.

Midpoint Calculation

Midpoint calculation involves finding the exact middle point of a line segment, joining two endpoints. This point is crucial in balancing or dividing lines equally.
How to Calculate the Midpoint

• Midpoint Formula: The formula \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \) allows you to calculate the midpoint of the segment from its endpoints \((x_1, y_1)\) and \((x_2, y_2)\). This formula is derived by averaging the x-coordinates and y-coordinates of the endpoints.
• Example Calculation: For the endpoints \((6, 0)\) and \((13, 0)\), the midpoint calculation is as follows:
  • First, average the x-coordinates: \(\frac{6 + 13}{2} = 9.5\).
  • Next, average the y-coordinates: \(\frac{0 + 0}{2} = 0\).
  • Thus, the midpoint is \((9.5, 0)\).

This midpoint at \((9.5, 0)\) is on the x-axis, reflecting that the y-coordinates were zero, simplifying our calculations. Midpoints are vital in various applications, such as finding the center of balance or dividing tasks evenly.