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Chapter 2: Problem 19

Find the midpoint of each line segment with the given endpoints. \(S(6,8) and(2,4)S\)

Short Answer

Expert verified
The midpoint of the line segment with endpoints \(S(6,8)\) and \(S(2,4)\) is \(M=(4,6)\).

Step by step solution

01

Identify the coordinates

Identify the x and y coordinates for each endpoint. Here, for point S we have \(x_1 = 6\) and \(y_1 = 8\), and for point S we have \(x_2 = 2\) and \(y_2 = 4\).
02

Apply the Midpoint Formula

Next, you apply the midpoint formula by adding the x-coordinates (\(x_1\) and \(x_2\)) of the given endpoints and dividing by 2 to find the x-coordinate of the midpoint. Do the same for the y-coordinates (\(y_1\) and \(y_2\)). The formulas are: \\[Midpoint_x = \frac{(x_1 + x_2)}{2}\\Midpoint_y = \frac{(y_1 + y_2)}{2}\] This will give you the x-coordinate and y-coordinate of the midpoint respectively. To get: \\[Midpoint_x = \frac{(6 + 2)}{2} = 4\\Midpoint_y = \frac{(8 + 4)}{2} = 6\]
03

Write the Midpoint

Formulate the midpoint using the resulting x and y coordinates. The midpoint is then written as \(M = (Midpoint_x, Midpoint_y)\). In this case, \(M =(4,6)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Coordinates
When you hear the term "coordinates," it simply refers to a pair of numbers that helps you determine a specific location on a plane. This plane is usually the Cartesian plane, which is like a grid with a horizontal x-axis and a vertical y-axis.
Think of each axis as a number line.
Coordinates are written as pairs between parentheses, such as \((x, y)\).
These numbers tell you how far along each axis the point is.
  • The "x" value tells you the horizontal position. If it's positive, the point is to the right; if negative, to the left.
  • The "y" value tells you the vertical position. If it's positive, the point is above the origin; if negative, below.
With our example points, S(6,8) and (2,4), the coordinates tell us exactly where these points exist in the plane.
What is a Line Segment?
A line segment is part of a line that connects two points directly. Think of any line that has endpoints; the section between those points is your line segment.
Unlike a line, a line segment has a definite start and end.
Imagine drawing a straight line with a ruler between point \((6,8)\) and \((2,4)\).
  • The endpoints of the segment are fixed at these two points.
  • No matter how long or short the segment is, it's defined solely by these endpoints.
This makes the midpoint of the segment simply the halfway point between the two endpoints, easy to calculate using the midpoint formula.
Using Algebra for Midpoint Calculations
Algebra is a branch of math that allows us to solve problems with letters and numbers. In this case, it's helping us find the midpoint between two points on our line segment.

For the midpoint calculation, we use the formula: \[Midpoint = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

Here's how it works using algebra:
  • We have the given endpoints S(6,8) and (2,4). Assign the coordinates \(x_1 = 6\), \(y_1 = 8\), \(x_2 = 2\), and \(y_2 = 4\).
  • Plug these values into the midpoint formula:
  • Compute the midpoint_x: \(\frac{6 + 2}{2} = 4\)
  • Compute the midpoint_y: \(\frac{8 + 4}{2} = 6\)
  • Now, the midpoint is written as \(M = (4, 6)\).
Algebra simply helps us perform these calculations easily and ensure that we're finding the exact center of our line segment.

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Most popular questions from this chapter

If a function is defined by an equation, explain how to find its domain.

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