Problem 97 - Algebra A Combined Function

Question

Find the midpoint of each line segment whose endpoints are given. (4.6,-3.5)\(;(7.8,-9.8)\)

Step-by-Step Solution

Answer

The midpoint is (6.2, -6.65).

Step 1: Understanding the Midpoint Formula

To find the midpoint of a line segment between two points \(x_1, y_1\) and \(x_2, y_2\), we use the midpoint formula \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). This formula gives the coordinates of the midpoint.

Step 2: Substituting the Given Values

For the given points \(4.6, -3.5\) and \(7.8, -9.8\), substitute these into the midpoint formula: \(x_1 = 4.6\), \(y_1 = -3.5\), \(x_2 = 7.8\), \(y_2 = -9.8\).

Step 3: Calculate the Midpoint X-Coordinate

First, calculate the x-coordinate of the midpoint: \[ \frac{x_1 + x_2}{2} = \frac{4.6 + 7.8}{2} = \frac{12.4}{2} = 6.2 \]

Step 4: Calculate the Midpoint Y-Coordinate

Next, calculate the y-coordinate of the midpoint: \[ \frac{y_1 + y_2}{2} = \frac{-3.5 + (-9.8)}{2} = \frac{-13.3}{2} = -6.65 \]

Step 5: Final Midpoint Coordinates

Combine the calculated x and y coordinates to get the midpoint: The midpoint of the line segment is \(6.2, -6.65\).

Key Concepts

Coordinates

Coordinates are essential in geometry for locating points on a plane. Each point on this geometric plane can be identified with a pair of numbers, known as coordinates. These are usually presented as \(x, y\), where \x\ represents the horizontal position and \y\ denotes the vertical position.
Using coordinates, we can easily describe the location of points in a simple and precise manner. For example, the endpoints of a line segment might be given as \(4.6, -3.5\) and \(7.8, -9.8\). Here, \(4.6, -3.5\) indicates a point located 4.6 units right along the x-axis and 3.5 units down along the y-axis. In the same way, \(7.8, -9.8\) signifies a point further along the plane.
Understanding how to work with coordinates is foundational for further concepts like finding midpoints or calculating distances. Knowing these basics allows us to manipulate and explore shapes and lines effectively.

Line segment

A line segment is a part of a line that is bounded by two distinct endpoints. Unlike a line which extends infinitely in both directions, a line segment has a finite length. It's helpful to think of it as a piece of string with two knotted ends and no extra length beyond those knots.
When dealing with line segments in math problems, you often need to consider their endpoints, which are given as coordinates. This is crucial when finding the midpoint of the segment, a common geometric calculation that identifies the exact center point between these two knots.
For example, if you have a line segment with endpoints at \(4.6, -3.5\) and \(7.8, -9.8\), you are dealing with a finite section of the line that begins exactly at one of these coordinates and ends precisely at the other. By understanding this concept, you can more easily grasp related tasks such as midpoint calculation.

Calculation

Calculation is at the heart of many geometry problems, and the midpoint formula is a perfect example. This formula is used to find the point that is exactly halfway between two given endpoints of a line segment. Given the equation, \M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\,\ you can determine the coordinates of this midpoint.
Let's break it down using our example of endpoints \(4.6, -3.5\) and \(7.8, -9.8\):

• First, to find the x-coordinate of the midpoint, add the x-coordinates of the endpoints and divide by 2: \frac{4.6 + 7.8}{2} = 6.2\.
• Then, find the y-coordinate by adding the y-coordinates and dividing by 2: \frac{-3.5 + (-9.8)}{2} = -6.65\.
• Finally, combine these two results to get the midpoint: \(6.2, -6.65\).

Using calculation in this way helps uncover the hidden details of the shapes and lines drawn using coordinates, directly linking numeric manipulation with geometric understanding.