Problem 19 - College Algebra Essentials

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 line segment with endpoints (6,8) and (2,4) is (4, 6).

Step 1: Identify the given points

The two given points are (6,8) and (2,4). We will identify (6,8) as point \(A (x_{1}, y_{1})\) and (2,4) as point \(B (x_{2}, y_{2})\). So, \(x_{1}= 6, y_{1}= 8, x_{2}= 2, y_{2}= 4.\)

Step 2: Substitute in the Midpoint Formula

We have the formula for the midpoint: \((\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2})\). Substituting the given points into the formula we get: \((\frac{6+2}{2}, \frac{8+4}{2})\).

Step 3: Compute the Midpoint

Evaluating the expressions in the parentheses we get: \((\frac{8}{2}, \frac{12}{2})\), which simplifies to: \((4, 6)\).

Key Concepts

Line Segments

In geometry, a **line segment** is a straight line which links two points without extending beyond them. Think of it as the shortest path connecting two coordinate positions on a graph. Unlike a line that stretches infinitely, a line segment has two endpoints.

• Endpoints are essential; they dictate the start and finish of a segment.
• We use points like (6,8) and (2,4) to define a line segment precisely.
• These endpoints help in calculating the midpoint, which is the exact center of the segment.

Understanding line segments is pivotal when dealing with geometry problems, as they form the basis for more complex shapes and calculations.

Coordinate Geometry

**Coordinate Geometry**, also known as analytic geometry, brings algebra and geometry together by using a coordinate plane. This method allows us to dive deeper into geometric shapes by leveraging algebra.

• Using equations and formulas, we can accurately calculate distances, slopes, and midpoints.
• The formula for the midpoint, for instance, helps in determining the center point between two endpoints of a line segment.

Utilizing coordinate geometry is helpful in visualizing shapes, sizes, and relationships within geometric figures, making complex problems more approachable.

Cartesian Plane

The **Cartesian Plane** is a two-dimensional surface defined by two intersecting perpendicular lines called axes: the horizontal x-axis and vertical y-axis.

• Coordinates, like (6,8) and (2,4), are used to specify exact positions on this plane.
• Both axes split the plane into four quadrants, each helping to determine the sign (+/-) and position of coordinates.

Understanding the Cartesian Plane is fundamental for solving problems involving two-dimensional geometry, especially when using formulas like the midpoint formula to discover precise points between known positions.