Problem 24 - College Algebra with Corequisite Support

Question

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points. $$(-5,-3) \text { and }(-2,-8)$$

Step-by-Step Solution

Answer

The midpoint is \((-3.5, -5.5)\).

Step 1: Understand the Problem

We need to find the midpoint of the line segment with endpoints \((-5,-3)\) and \((-2,-8)\). The midpoint formula will be used to solve the problem.

Step 2: Apply the Midpoint Formula

The midpoint \( M \) of a line segment joining the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Substitute \(x_1 = -5\), \(y_1 = -3\), \(x_2 = -2\), and \(y_2 = -8\) into the formula.

Step 3: Calculate the Midpoint's x-coordinate

Use the x-coordinates of the given points: \(-5\) and \(-2\). Calculate the x-coordinate of the midpoint:\[ x = \frac{-5 + (-2)}{2} = \frac{-7}{2} = -3.5 \]

Step 4: Calculate the Midpoint's y-coordinate

Use the y-coordinates of the given points: \(-3\) and \(-8\). Calculate the y-coordinate of the midpoint:\[ y = \frac{-3 + (-8)}{2} = \frac{-11}{2} = -5.5 \]

Step 5: Combine the Results

The midpoint of the line segment is the combination of the x-coordinate and y-coordinate results: \((-3.5, -5.5)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra and geometric figures through the coordinate plane. In a coordinate plane, points are defined by pairs of numbers known as coordinates. These coordinates help in locating points, lines, and even more complex shapes.

• A point is represented by \( (x, y) \) where \( x \) is the horizontal component and \( y \) is the vertical component.
• By using coordinate geometry, we can perform precise calculations, such as finding the midpoint between two points or calculating the distance of a line segment.

In this field, it's crucial to understand how algebraic formulas can help solve geometric problems.

Line Segment

A line segment is a part of a line that has two endpoints. It includes every point between these endpoints. Unlike a line that extends infinitely in both directions, a segment is finite.

• The endpoints represent the positions where the line segment starts and ends on a coordinate plane.
• For example, in our exercise, the points \( (-5, -3) \) and \( (-2, -8) \) are the endpoints of our line segment.

When dealing with such a line segment, it's often useful to find its midpoint, which divides the segment into two equal parts.

Midpoint Calculation

The midpoint is the point exactly halfway between two other points on a line segment. Calculating the midpoint involves using the midpoint formula. This formula averages the \( x \) and \( y \) coordinates of the endpoints.

Understanding the Midpoint Formula

The formula is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

• Plug in the \( x \) values of the endpoints into the formula: \( \left( -5, -3 \right) \) and \( \left( -2, -8 \right) \).
• Do the same with the \( y \) values. This results in the midpoint \( (-3.5, -5.5) \).

This process provides a clear and concise way to find the middle of a line segment.

Algebra Problem Solving

Algebra problem solving often involves using formulas to find unknown values. In geometry, these solutions frequently involve plugging numbers into known equations.

• The midpoint formula is an example, serving as a practical tool within algebra and geometry.
• Substitute the given coordinates into the formula: first calculate the sum of the \( x \)-coordinates and divide by 2, then do the same for the \( y \)-coordinates.

Step-by-step calculations simplify the problem, helping you arrive at accurate results. Algebra reduces complex geometric problems into manageable arithmetic tasks, boosting overall problem-solving skills.