Problem 19 - College Algebra

Question

Find the midpoint of each line segment with the given endpoints. $$(6,8) \text { and }(2,4)$$

Step-by-Step Solution

Answer

The midpoint of the given line segment with endpoints (6,8) and (2,4) is (4,6).

Step 1: Identify the coordinates

Firstly, identify the coordinates of the endpoints. The given endpoints of the line segment are (6,8) and (2,4). Therefore, we have x1 = 6, y1 = 8, x2 = 2, y2 = 4.

Step 2: Find the x-coordinate of the midpoint

After identifying the coordinates, find the x-coordinate of the midpoint. According to the formula, the x-coordinate of the midpoint = \( \frac{{x1 + x2}}{2} \) = \( \frac{{6 + 2}}{2} \) = 4.

Step 3: Find the y-coordinate of the midpoint

Similar to step 2, now find the y-coordinate of the midpoint. The y-coordinate of the midpoint = \( \frac{{y1 + y2}}{2} \) = \( \frac{{8 + 4}}{2} \) = 6.

Step 4: Write down the coordinates of the midpoint

Finally, express the coordinates of the midpoint in the (x, y) format, and so the midpoint is (4, 6).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric figures and understand their properties. Within this system, points on the plane are represented using coordinates. Each point in a plane has a unique pair of numbers, denoted as \(x, y\). These numbers correspond to the horizontal (x-axis) and vertical (y-axis) distances from a fixed point called the origin. This framework allows us to analyze the features of shapes and solve various geometric problems using equations.

• Coordinates make it possible to describe the location of points precisely.
• The Cartesian plane divides the space into quadrants, providing a useful reference for locating and visualizing points.
• Coordinate geometry bridges the gap between algebra and geometry, offering a powerful toolset for solving complex problems.

Understanding coordinate geometry is essential for calculating midpoints, which involves the relationship between points on the plane and how they form various geometric figures like lines and line segments.

Line Segments

A line segment is a part of a line that has two endpoints, making it a fundamental concept in geometry. Unlike lines that extend indefinitely, line segments have a finite length and are determined by their two endpoints. For the exercise, we are interested in the line segment formed by the points (6,8) and (2,4).

• The length of a line segment can be measured using various distance formulas derived from coordinate geometry.
• Line segments can lie in any direction on the plane, defined by their endpoints' coordinates.
• Understanding line segments is crucial for breaking down more complex shapes into simpler, manageable parts.

Learning about line segments provides a foundational understanding that is applicable in more advanced topics like vectors, polygons, and more. By analyzing endpoints, we can determine key properties like midpoints, slopes, and lengths, making line segments a versatile element in mathematical studies.

Calculation of Midpoints

The calculation of midpoints is a straightforward yet vital procedure in coordinate geometry. The midpoint of a line segment is the point that divides the segment into two equal parts. It is calculated by averaging the coordinates of the endpoints, providing a central point on the segment.
To calculate the midpoint, use the formula:

• Midpoint \(M(x, y) = \frac{{((x_1 + x_2))}}{2}, \frac{{((y_1 + y_2))}}{2}\)
• This requires simply finding the average of the x-coordinates and the y-coordinates of the endpoints.

In our exercise, the endpoints are (6,8) and (2,4):

• For the x-coordinate, compute \( \frac{{6 + 2}}{2} = 4\)
• For the y-coordinate, compute \( \frac{{8 + 4}}{2} = 6\)

Thus, the midpoint coordinates are (4, 6).
The concept of midpoints is widely used not only in basic geometry but also in real-world applications such as engineering, architecture, and even graphic design, providing a balanced and symmetrical view of a line segment.