Problem 38 - College Algebra with Modeling and Visualization

Question

Find the midpoint of the line segment connecting the points. $$ (-6,7),(9,-4) $$

Step-by-Step Solution

Answer

The midpoint is (1.5, 1.5).

Step 1: Understand the midpoint formula

The midpoint of a line segment is the point that divides the segment into two equal parts. The formula to find the midpoint \((M)\) of a line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].

Step 2: Identify the coordinates of the endpoints

We are given two points: \((-6, 7)\) and \((9, -4)\). Here, \(x_1 = -6\), \(y_1 = 7\), \(x_2 = 9\), and \(y_2 = -4\).

Step 3: Apply the formula to find the x-coordinate of the midpoint

Substitute \(x_1\) and \(x_2\) into the x-part of the midpoint formula: \[ \frac{x_1 + x_2}{2} = \frac{-6 + 9}{2} = \frac{3}{2} = 1.5 \].

Step 4: Apply the formula to find the y-coordinate of the midpoint

Substitute \(y_1\) and \(y_2\) into the y-part of the midpoint formula: \[ \frac{y_1 + y_2}{2} = \frac{7 + (-4)}{2} = \frac{3}{2} = 1.5 \].

Step 5: Write the coordinates of the midpoint

Combine the results from Steps 3 and 4. Thus, the midpoint \( M \) is \((1.5, 1.5)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is where algebra meets geometry to explore relationships between figures in a coordinate plane. This incredible branch of mathematics allows us to use a coordinate system, typically denoted by a grid of horizontal (x-axis) and vertical (y-axis) lines, to precisely describe and analyze the positions and relationships of geometric figures.

Each point on this grid has a unique set of coordinates, typically expressed as \( (x, y) \), which indicate its position relative to the axes.

• The x-coordinate specifies horizontal movement from the origin (the point where axes intersect, usually denoted as \( (0,0) \)).
• The y-coordinate describes vertical movement from the origin.

In context of our problem, coordinate geometry helps us find the midpoint along the line segment that connects two specified points in this plane. Understanding relationships through coordinate geometry simplifies complex geometric problems by transforming them into more manageable algebraic tasks.

Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints. Unlike a line that stretches infinitely in both directions, a line segment is finite and defined solely by these endpoints. Line segments are fundamental elements in geometry and play an especially important role in coordinate geometry for defining distances, directions, and midpoints.

For instance, when dealing with the given endpoints \( (-6, 7) \) and \( (9, -4) \), they create a specific segment within the coordinate plane.

• This specific segment allows us to explore geometric concepts like distance between points, slope, and particularly the midpoint – which is the focus of our exercise.
• The midpoint divides the segment into two equal halves, making its understanding essential in analyzing geometrical figures and solving spatial problems.

Thus, every line segment in the coordinate plane not only represents a geometric object but also ties into analytical exploration through formulas like distance and midpoint formulas.

Mathematical Calculation

Mathematical calculations are the foundation of problem solving in coordinate geometry. This process involves using well-established formulas to handle and simplify geometric problems. For our problem, the calculation process is primarily concerned with using the Midpoint Formula. The formula is given as: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].

This formula is applied in the following steps:

• Identify and substitute the coordinates of the endpoints into the formula. For our case: \( (-6, 7) \) and \( (9, -4) \).
• Calculate the average of the x-coordinates: \( \frac{-6 + 9}{2} = 1.5 \).
• Calculate the average of the y-coordinates: \( \frac{7 + (-4)}{2} = 1.5 \).
• Finally, combine these results to give the midpoint coordinates, \( (1.5, 1.5) \).

This calculation tells us exactly where the middle point of the segment lies. Being able to perform mathematical calculations like this, with a step-by-step methodology, is crucial for accurately solving and understanding many geometry problems.