Problem 41 - College Algebra with Modeling and Visualization

Question

Find the midpoint of the line segment connecting the points. $$ (1.5,2.9),(-5.7,-3.6) $$

Step-by-Step Solution

Answer

The midpoint is (-2.1, -0.35).

Step 1: Identify the coordinates

Identify the coordinates of the two points that form the endpoints of the line segment. Here, the coordinates are \((x_1, y_1) = (1.5, 2.9)\) and \((x_2, y_2) = (-5.7, -3.6)\).

Step 2: Use the midpoint formula

The formula to find the midpoint \((x_m, y_m)\) of a line segment with given endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2} \]

Step 3: Calculate the midpoint's x-coordinate

Substitute the x-coordinates of the given points into the midpoint formula to find the x-coordinate of the midpoint: \[ x_m = \frac{1.5 + (-5.7)}{2} = \frac{1.5 - 5.7}{2} = \frac{-4.2}{2} = -2.1 \]

Step 4: Calculate the midpoint's y-coordinate

Substitute the y-coordinates of the given points into the midpoint formula to find the y-coordinate of the midpoint: \[ y_m = \frac{2.9 + (-3.6)}{2} = \frac{2.9 - 3.6}{2} = \frac{-0.7}{2} = -0.35 \]

Step 5: Write down the midpoint coordinates

Combine the calculated x and y values to find the coordinates of the midpoint of the line segment. Therefore, the midpoint is \((-2.1, -0.35)\).

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra with geometry to visualize points, lines, and curves on a two-dimensional plane. This branch of geometry allows us to describe the positions of points through pairs of numbers, known as coordinates. In coordinate geometry, the horizontal and vertical axes, typically labeled as the x-axis and y-axis, form a grid. Any point on this grid is identified using an ordered pair \(x, y\). This mix of numerical and graphical representation is key in solving geometric problems efficiently.
By applying algebra to geometry, we can derive formulas that calculate specific properties like distances, midpoints, and slopes of lines. For example, the midpoint formula helps find the center of a line segment, giving us valuable insights about its position within the coordinate plane.

Exploring Line Segments

A line segment is a fundamental entity in geometry that represents the shortest path connecting two points. Unlike a line, which extends infinitely in both directions, a line segment has a fixed length, defined by its two endpoints. Think of it as a piece of a line, cut and held between these two points. When discussing the midpoint of a line segment, we look for a point that is equidistant from both endpoints, effectively splitting the segment into two equal halves.
By finding the midpoint, we gain a precise understanding of the line segment's position in the coordinate plane. This process is crucial in many real-world applications, such as computer graphics, physics, and engineering, where spatial accuracy is key.

The Role of Algebra in Geometry

Algebra plays a pivotal role in geometry, providing a framework to describe and solve geometric problems using equations and formulas. By using algebraic techniques, we convert geometric relations into algebraic expressions, making it easier to perform calculations and derive results.
The midpoint formula is a perfect example of algebra's utility in geometry. The formula \((x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)\ wraps the idea of a halfway point in a line segment into a simple, calculated form.

• This formula lets us quickly and accurately determine the specific location of the midpoint.
• Algebraic manipulation, like adding or dividing coordinates, breaks down complex spatial problems into manageable parts.

Through these methods, algebra enables more efficient and precise exploration of geometric concepts.