Problem 62 - Elementary and Intermediate Algebra

Question

Find the midpoint of the line segment with the given endpoints. \((-4,-3),(4,-8)\)

Step-by-Step Solution

Answer

The midpoint is \((0, -5.5)\).

Step 1: Identify the Coordinates

First, identify the coordinates of the given endpoints. We have:- Endpoint A: - x-coordinate: \(-4\) - y-coordinate: \(-3\)- Endpoint B: - x-coordinate: \(4\) - y-coordinate: \(-8\)

Step 2: Use the Midpoint Formula

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

Step 3: Calculate the x-coordinate of the Midpoint

Substitute the x-coordinates of the endpoints into the formula:\[x_m = \frac{-4 + 4}{2} = \frac{0}{2} = 0\]

Step 4: Calculate the y-coordinate of the Midpoint

Substitute the y-coordinates of the endpoints into the formula:\[y_m = \frac{-3 + (-8)}{2} = \frac{-11}{2} = -5.5\]

Step 5: Write the Midpoint Coordinates

The midpoint coordinates are:\((x_m, y_m) = (0, -5.5)\)

Key Concepts

Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints and contains every point on the line between its endpoints. Imagine it as a piece of a string stretched straight between two fixed points. Line segments are different from lines, which continue infinitely in both directions. In geometry, identifying the exact position of points along a line segment is crucial for understanding various properties, like the midpoint or length of the segment.

• A line segment has a specific length, which is the distance between its two endpoints.
• Unlike a line, a line segment does not go on forever; it has clear start and end points.
• The endpoints can be anywhere on a plane or in space, hence having coordinates marked as \(x, y\) in 2D or \(x, y, z\) in 3D.

Understanding line segments is foundational for calculating the midpoint, which divides the segment into two equal halves.

Coordinates

Coordinates are numbers that define the position of a point in a space. They are used in geometry to determine exact locations on a plane, allowing for precise plotting and mathematical computations.

• In two-dimensional space, coordinates are presented as ordered pairs \(x, y\), where \(x\) represents the horizontal position, and \(y\) represents the vertical position.
• Coordinates make it possible to perform various geometric operations, such as finding distances, angles, or midpoints between points.

To find the midpoint of a line segment, it's essential to know the coordinates of its endpoints. For example, in the exercise, the endpoints are given as \((-4, -3)\) and \(4, -8\). Knowing these coordinates allows us to plug them into mathematical formulas for calculations like midpoint calculation.

Midpoint Calculation

Midpoint calculation involves finding the point that lies halfway between two endpoints of a line segment. It is like finding the average position between two points. The midpoint is an essential concept in both geometry and algebra because it helps in dividing a segment into two equal parts.To calculate the midpoint, we use the

Midpoint Formula

The formula for finding the midpoint \(M(x_m, y_m)\) for endpoints \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is:\[x_m = \frac{x_1 + x_2}{2}\] \[y_m = \frac{y_1 + y_2}{2}\] In our example:

• Substitute in: \(x_1 = -4\), \(y_1 = -3\), \(x_2 = 4\), and \(y_2 = -8\).
• Calculate \(x_m\): \(x_m = \frac{-4 + 4}{2} = 0\).
• Calculate \(y_m\): \(y_m = \frac{-3 + (-8)}{2} = -5.5\).
• Thus, the midpoint is \((0, -5.5)\).

By following these steps, you can ensure accurate midpoint calculations for any line segment with given endpoints.