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Chapter 2: Problem 24

Find the midpoint of each line segment with the given endpoints. $$(-2,-1)\( and \)(-8,6)$$

Short Answer

Expert verified
The midpoint of the line segment with endpoints \((-2,-1)\) and \((-8,6)\) is \((-5, 2.5)\)

Step by step solution

01

Identify the coordinates of the endpoints

Recognize the given points as \((-2, -1)\) and \((-8, 6)\). These correspond to \((x_1, y_1)\) and \((x_2, y_2)\) respectively in the midpoint formula.
02

Apply the Midpoint formula

Plugging the coordinates we identified in step 1 into the midpoint formula, we can find the midpoint of the line segment. The x-coordinate of the midpoint is \((x_1 + x_2)/2 = (-2 - 8)/2 = -10/2 = -5\). Similarly, the y-coordinate of the midpoint is \((y_1 + y_2)/2 = (-1 + 6)/2 = 5/2 = 2.5\).
03

Write the Midpoint

Combine the x and y coordinates from step 2 to form the coordinates of the midpoint, giving us \((-5, 2.5)\).

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Key Concepts

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

Understanding Line Segments
A line segment is a part of a line that has two endpoints. Unlike a line, which stretches infinitely in both directions, a line segment has a start and an end. In our exercise, the endpoints are given as coordinates, specifically
  • \((-2, -1)\)
  • \((-8, 6)\)
These points define the boundary of the line segment. You can think of a line segment as a "cut piece" of a line. It鈥檚 like taking a string and cutting it to a specific length. This concept is crucial in geometry and helps us navigate the notion of distance and position on a plane. Understanding a line segment helps to find midpoints, distances, and other geometric properties.
What Coordinates Communicate
Coordinates are pairs of numbers used to determine the position of a point on a plane. There are two values: the first is the x-coordinate, and the second is the y-coordinate. They are usually written in parentheses, like
  • \((x, y)\)
Coordinates are like addresses for points in the plane. They inform us precisely where a point lies in relation to the two reference axes, the x-axis, and the y-axis. In our example, the points \((-2, -1)\) and \((-8, 6)\) tell us the exact location of the endpoints of our line segment on a Cartesian plane. With this information, we can easily locate and plot the points, helping us in calculations like finding the midpoint.
Exploring the X-Coordinate
The x-coordinate is the first number in an ordered pair and tells us how far left or right a point is along the x-axis. It represents the horizontal position. A positive x-value indicates the point is to the right of the origin, while a negative value means it's to the left. For our points:
  • \((-2, -1)\) - here the x-coordinate is \(-2\). This tells us the point is two units to the left on the x-axis.
  • \((-8, 6)\) - with an x-coordinate of \(-8\), indicating it's eight units left of the origin.
When we compute the midpoint, we average these x-coordinates to find a middle position horizontally between the two endpoints. This balancing of positions is essential for accurately locating the midpoint.
Grasping the Y-Coordinate
The y-coordinate is the second value in an ordered pair, indicating how far up or down a point is on the y-axis. It's all about vertical placement. Positive y-values mean the point is above the x-axis, and negative values show it's below. In the current problem:
  • \((-2, -1)\) - here, the y-coordinate is \(-1\), placing it one unit below the x-axis.
  • \((-8, 6)\) - has a y-coordinate of \(6\), positioning it six units above the x-axis.
Calculating the midpoint involves finding the average of these y-values, which indicates the central vertical position between the two endpoints. This ensures that our midpoint truly sits at the middle in terms of vertical distance as well.

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Most popular questions from this chapter

Here is the 2011 Federal Tax Rate Schedule \(X\) that specifies the tax owed by a single taxpayer. (TABLE CAN'T COPY) The preceding tax table can be modeled by a piecewise function, where \(x\) represents the taxable income of a single taxpayer and \(T(x)\) is the tax owed: $$T(x)=\left\\{\begin{array}{c}0.10 x \\\850.00+0.15(x-8500) \\\4750.00+0.25(x-34,500) \\\17,025.00+0.28(x-83,600) \\\\\frac{?}{?}\end{array}\right.$$ if \(\quad 0 < x \leq 8500\) if \(\quad 8500 < x \leq 34,500\) if \(\quad 34,500 < x \approx 83,600\) if \(\quad 83,600 < x =174,400\) if \(174,400 < x \leq 379,150\) if \(\quad x >379,150\) Use this information to solve. Find and interpret \(T(50,000)\).

Assume that \((a, b)\) is a point on the graph of \(f .\) What is the corresponding point on the graph of each of the following functions? $$ y-f(x-3) $$

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)-x^{3}-3 $$

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)-2|x+4| $$

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$g(x)=x^{\frac{2}{3}}$$
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