Problem 22 - College Algebra

Question

Find the midpoint of each line segment with the given endpoints. $$(-4,-7)\( and \)(-1,-3)$$

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints (-4,-7) and (-1,-3) is (-2.5, -5).

Step 1: Identify the Coordinates

First, identify the coordinates of the two given points. The first point is (-4, -7) so x1 = -4 and y1 = -7. The second point is (-1, -3) so x2 = -1 and y2 = -3.

Step 2: Use the Midpoint Formula

Now, use the formula for finding the midpoint: [(x1 + x2) / 2, (y1 + y2) / 2]. Applying the formula gives the following operations: ((-4 + -1) / 2, (-7 + -3) / 2).

Step 3: Simplify the Expressions

Simplify the expressions to find the coordinates of the midpoint. The result would be (-5 / 2, -10 / 2) after summing the coordinates and dividing each by 2.

Step 4: Simplify to Decimal

Simplify each coordinate to a decimal to find the exact location of the midpoint. The result will be as follows: (-2.5, -5). This is the midpoint of the line segment.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a fascinating branch of mathematics that ties together algebra and geometry. It involves using a coordinate system to study geometric problems. When we talk about coordinates, we're talking about ordered pairs of numbers that represent points on a grid, typically a two-dimensional plane. The point \((-4,-7)\) denotes moving 4 units left and 7 units down from the origin on this plane.
Moving to another point, like \((-1,-3)\), involves a different set of movements along our grid.
Understanding the relevance of these coordinates means recognizing how each point is defined by its distance from two perpendicular axes:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically.

The interplay between these axes and points is key when solving problems about line segments, like finding midpoints. Being equipped with knowledge of coordinate geometry helps in visualizing and solving these problems more effectively.

Algebraic Formulas

Algebraic formulas serve as tools for manipulating numerical relationships in mathematics. In coordinate geometry, one particularly useful formula is the Midpoint Formula, used to find a point exactly halfway between two other points.
The formula is:\[ (x_m, y_m) = rac{{(x_1 + x_2)}}{2}, rac{{(y_1 + y_2)}}{2 } \]This formula calculates the average of the x-coordinates and the y-coordinates separately, giving two numbers which together become the midpoint coordinates:

• The x-component of the midpoint: \( rac{{x_1 + x_2}}{2} \)
• The y-component of the midpoint: \( rac{{y_1 + y_2}}{2} \)

Using these components, students can see how algebra easily helps in determining precise locations in coordinate geometry.

Problem Solving Steps

Approaching problems systematically is crucial, especially in mathematics.Begin by clearly identifying what is being asked, like finding the midpoint of a line segment.
By employing structured problem-solving steps:

• Identify coordinates of endpoints: Knowing the exact points you start with, e.g., (-4, -7) and (-1, -3).
• Apply the formula: Use \( rac{{(x_1 + x_2)}}{2} \) and \( rac{{(y_1 + y_2)}}{2} \) to calculate the midpoint.
• Simplify: Execute the mathematics to simplify any fractions or convert to decimals, like turning-5/2 into -2.5.

Following these steps provides a clear pathway to the answer and enhances understanding by breaking down the problem into manageable parts. Community exposure to these methodical steps boosts problem-solving skills, cultivates logical reasoning, and bridges conceptual gaps for students.