Problem 26 - College Algebra Essentials

Question

find the midpoint of each line segment with the given endpoints. $$ \left(-\frac{2}{5}, \frac{7}{515}\right) \text { and }\left(-\frac{2}{5},-\frac{4}{15}\right) $$

Step-by-Step Solution

Answer

The midpoint between the points \((-\frac{2}{5}, \frac{7}{515})\) and \((-\frac{2}{5}, -\frac{4}{15})\) is \((-\frac{2}{5}, \frac{-127}{3090})\).

Step 1: Identify the coordinates

First, identify the given coordinates. They are \((-\frac{2}{5}, \frac{7}{515})\) and \((-\frac{2}{5}, -\frac{4}{15})\). The coordinates for the first point are \(x_1 = -\frac{2}{5}\) and \(y_1 = \frac{7}{515}\), the coordinates for the second point are \(x_2 = -\frac{2}{5}\) and \(y_2 = -\frac{4}{15}\).

Step 2: Apply the midpoint formula

Now, apply the midpoint formula to find the midpoint of the two points. The formula is \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\). Substituting the given coordinates into this formula, we get \(\left(\frac{-\frac{2}{5} + -\frac{2}{5}}{2}, \frac{\frac{7}{515} + -\frac{4}{15}}{2}\right)\).

Step 3: Simplify the midpoint

Next, simplify the calculation. The x-coordinate is \(\frac{-\frac{2}{5} + -\frac{2}{5}}{2} = -\frac{2}{5}\). The y-coordinate is \(\frac{\frac{7}{515} + -\frac{4}{15}}{2}=\frac{-127}{3090}\). So the midpoint is \(\left(-\frac{2}{5}, \frac{-127}{3090} \right)\).

Key Concepts

Coordinate Geometry

In coordinate geometry, often referred to as analytical geometry, we learn about points, lines, and shapes on the Cartesian plane. The Cartesian plane is a two-dimensional surface defined by horizontal and vertical axes, usually labeled as the x-axis and y-axis. The location of any point on this plane is described by an ordered pair of numbers (x, y), known as coordinates.

For example, the coordinates \(\left(-\frac{2}{5}, \frac{7}{515}\right)\) and \(\left(-\frac{2}{5}, -\frac{4}{15}\right)\) from the exercise represent specific locations on the Cartesian plane. These points are plotted based on their distance from the origin (0, 0) along the x and y axes. Furthermore, the concept of the midpoint is a core aspect of coordinate geometry. The midpoint of a line segment is the point that is exactly midway between the endpoints of the segment. It's important not just in theory but in various applications like computer graphics, navigation, and engineering designs.

Algebraic Fractions

Algebraic fractions are fractions that contain algebraic expressions in their numerators, denominators, or both. They are like regular fractions, but instead of just numbers, they include variables and can express more complex relationships. Simplifying algebraic fractions is an essential skill in mathematics as it is frequently used in various areas of algebra including solving equations and dealing with functions.

Simplifying the algebraic fractions in the coordinates from the exercise, such as \(\frac{7}{515} + -\frac{4}{15}\), involves finding a common denominator and combining the numerators appropriately, followed by reducing the fraction to its simplest form. Understanding how to work with algebraic fractions is crucial for higher-level mathematics, including calculus and beyond.

Math Problem Solving

Math problem solving is a process that involves understanding a problem, devising a plan, carrying out the plan, and finally evaluating the solution for accuracy. In the context of our exercise – finding the midpoint – it's important to proceed step by step. First, one identifies the given information, which in this case are the coordinates of two endpoints.

Next, a mathematical model or formula is applied; for midpoints, we use the midpoint formula. After substituting the specifics of the problem into the formula, the focus then shifts to simplification and calculation. This is where algebraic skills come into play, as simplifying the coordinates involves operations with algebraic fractions. Finally, one should review the solution to ensure it makes sense both numerically and within the context of the problem, in this case, ensuring the midpoint lies between the given endpoints on the Cartesian plane.