Problem 26 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

So, the midpoint of the given line segment is \(\left(-\frac{2}{5},\frac{1}{10}\right)\).

Step 1: Identify the coordinates

First, one should identify the coordinates for each endpoint: for the first point, \(x_1=-\frac{2}{5}\) and \(y_1=\frac{7}{15}\), and for the second point \(x_2=-\frac{2}{5}\) and \(y_2=-\frac{4}{15}\).

Step 2: Calculate the x-coordinate of the midpoint

The midpoint's x-coordinate is given by \(\frac{x_1 + x_2}{2}\). Substituting the given x-coordinates results in \(\frac{-\frac{2}{5} -\frac{2}{5}}{2} = -\frac{2}{5}\). Thus, the x-coordinate of the midpoint is \(-\frac{2}{5}\).

Step 3: Calculate the y-coordinate of the midpoint

The midpoint's y-coordinate is given by \(\frac{y_1 + y_2}{2}\). Substituting the given y-coordinates results in \(\frac{\frac{7}{15} - \frac{4}{15}}{2} = \frac{1}{10}\). Thus, the y-coordinate of the midpoint is \(\frac{1}{10}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a fascinating field that integrates algebra with geometry. It uses a coordinate system to describe positions on a plane. In this system, we use two numbers, often represented as \(x\) for the horizontal axis and \(y\) for the vertical axis, to denote the position of points in space.
By locating points using coordinates, we can analyze the geometry and algebraic relationships between them. This enables us to solve problems related to distance, slope, and midpoint of line segments. The coordinate system helps in understanding the layout of different geometric shapes and figures.

• Points: Defined by coordinates \(x\) and \(y\).
• Axes: Number lines perpendicular to each other (horizontal and vertical).
• Origin: Where both axes meet, i.e., \( (0,0) \).

Using coordinate geometry's tools, one can not just plot points but also find relationships between them, such as the midpoint which reveals the center point between two ends in a line segment.

Line Segments

A line segment is a part of a line that is defined by two endpoints. Unlike a line that extends infinitely in both directions, a line segment has a definite start and end. When working with line segments in coordinate geometry, each endpoint will be represented with a specific set of coordinates like \( (x_1, y_1) \) and \( (x_2, y_2) \).
Understanding the concept of line segments is crucial because they are the building blocks of many geometric constructs. In our exercise, the endpoints of the line segment were given as \( (-\frac{2}{5}, \frac{7}{15}) \) and \( (-\frac{2}{5}, -\frac{4}{15}) \).

• Every line segment can be part of geometric shapes like triangles and quadrilaterals.
• Length can be calculated using the distance formula.

Manipulating line segments and understanding their properties is essential in geometry and helps solve practical problems in design and construction.

Calculating Midpoints

Calculating the midpoint of a line segment in coordinate geometry is an important skill. The midpoint formula is used to find the point that is equidistant from the endpoints of a line segment. This formula is expressed as:
\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
Here, the coordinates \(x_1, y_1\) and \(x_2, y_2\) are the endpoints of the line segment. By plugging these coordinates into the midpoint formula, you determine the average of the x-coordinates and the average of the y-coordinates separately.

• The x-coordinate of the midpoint in our example was found to be \(-\frac{2}{5}\).
• The y-coordinate was found to be \(\frac{1}{10}\).
• The result gives the exact central point of the line segment.

This calculated point, the midpoint, is very useful in geometry, aiding in dividing lines, creating bisectors, and in symmetry operations. By mastering the midpoint formula, one gains a deeper understanding of balance and allocation of space in geometric contexts.