Problem 24 - College Algebra Essentials

Question

find the midpoint of each line segment with the given endpoints. $$ (-2,-1) \text { and }(-8,6) $$

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints (-2,-1) and (-8,6) is (-5, 2.5).

Step 1: Identify the coordinates

From the problem, we've got two points: Point A as (-2,-1) and Point B as (-8,6). Thus, the coordinates of point A are \(x_1 = -2\) and \(y_1 = -1\), while the coordinates of point B are \(x_2 = -8\) and \(y_2 = 6\).

Step 2: Apply the midpoint formula

Now, substitute \(x_1, y_1, x_2, y_2\) into the midpoint formula: \[ M = (\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}) \] This then results into: \[ M = (\frac{{-2 + (-8)}}{2}, \frac{{-1 + 6}}{2}) \]

Step 3: Simplify the formula

Simplifying the formula gives us \[ M = (\frac{{-10}}{2}, \frac{{5}}{2}) \] Further simplifying gives us the midpoint as \[ M = (-5, 2.5) \]

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a fascinating field of mathematics. It involves studying geometry through a coordinate system. The beauty of coordinate geometry is that it allows us to use algebraic techniques to solve geometric problems. For students, this branch of mathematics can be a bridge that links pure geometric concepts and algebra.

In our exercise, we are dealing with a line segment in the Cartesian plane. By identifying points on the plane using coordinates, we gain the ability to calculate distances, slopes, and midpoints analytically. In this scenario, the coordinates (-2,-1) and (-8,6) give specific locations on the plane. Understanding these positions and how they relate to each other is key to solving problems that involve determining midpoints and other geometrical properties.

• Coordinates are ordered pairs (x, y).
• Each point on the plane corresponds to a unique coordinate pair.
• Using these points, we can understand geometric constructions through calculations.

By combining these coordinates with algebraic methods like the midpoint formula, we deepen our grasp of geometry and develop problem-solving skills.

Algebraic Problem Solving

Algebraic problem solving is a fundamental skill in mathematics. It allows us to use expressions and formulas to arrive at solutions logically. When finding the midpoint of a line segment, we use algebra to substitute values into formulas. This is a systematic way to solve problems by converting them to an equation format.

The midpoint formula is expressed as \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
This formula takes the average of the x-coordinates and y-coordinates of the two endpoints. By substituting \( x_1 = -2, y_1 = -1, x_2 = -8, y_2 = 6 \), we perform a simple arithmetic calculation to locate the midpoint of the segment.

• Substitution of known values into formulas is key.
• Ensures accuracy in computation.
• Logical step-by-step process.

Applying these principles isn't only about getting the right answer; it's about understanding the algebraic structures behind seemingly straightforward calculations.

Line Segment

A line segment is a part of a line that connects two endpoints. Unlike a line, which extends indefinitely, a segment has a fixed length. In this exercise, we are interested in the specific section of a line that runs from one given point to another, in this case, from (-2,-1) to (-8,6).

Understanding a line segment is crucial because it enables us to make practical use of geometrical concepts. The task of finding a midpoint involves not just calculating an average position, but also understanding how these positions exist in space.

• Every line segment has a start and an end.
• Defined by two endpoints in coordinate geometry.
• Midpoint is the exact middle point, equidistant from both ends.

Visualizing this in the real world might involve imagining cutting a piece of string at its halfway point. You get a sense of balance and symmetry, properties that are central to geometry, and are nicely highlighted when working with midpoints specifically in line segments.