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Chapter 3: Problem 109

Find the midpoint of the line segment connecting the points (-2,1) and \(\left(\frac{3}{5},-4\right)\)

Short Answer

Expert verified
The midpoint is \( \left( \frac{-7}{10}, \frac{-3}{2} \right) \)

Step by step solution

01

Understand the midpoint formula

The midpoint of a line segment connecting two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
02

Identify the coordinates

Identify the given coordinates: \( x_1 = -2, y_1 = 1 \) and \( x_2 = \frac{3}{5}, y_2 = -4 \)
03

Apply the midpoint formula

Plug the coordinates into the midpoint formula: \[\text{Midpoint} = \left( \frac{-2 + \frac{3}{5}}{2}, \frac{1 + (-4)}{2} \right) \]
04

Simplify the expressions

Simplify each part of the midpoint formula: \[ \frac{-2 + \frac{3}{5}}{2} = \frac{-10 + 3}{10} = \frac{-7}{10} \] and \[ \frac{1 - 4}{2} = \frac{-3}{2} \]
05

Write the midpoint

Combine the simplified results to write the coordinates of the midpoint: \[ \text{Midpoint} = \left( \frac{-7}{10}, \frac{-3}{2} \right) \]

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Key Concepts

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

coordinates
To find the midpoint of a line segment, it's essential to understand the concept of coordinates. In a 2D space, a point is defined by two numbers known as coordinates, written as \( (x, y) \). The first number \( x \) represents the position on the horizontal axis, while the second number \( y \) represents the position on the vertical axis.
For example, in our given problem, the points are \((-2, 1)\) and \(\frac{3}{5}, -4\). Here, \(-2\) and \(\frac{3}{5}\) are the x-coordinates, and \(1\) and \(-4\) are the y-coordinates respectively.
simplifying expressions
When solving for the midpoint, you'll need to simplify mathematical expressions.
For example, let's take \(\frac{-2 + \frac{3}{5}}{2}\).
First, convert the expressions to have a common denominator: \(-2\) becomes \(\frac{-10}{5}\)
Then, you can add the fractions: \(-10 + 3 = -7\)
So the expression simplifies to \(\frac{-7}{10}\)
Similarly, for the second part: \frac{1 - 4}{2} = \frac{-3}{2}\
Properly simplifying these expressions helps in finding the accurate midpoint coordinates.
line segment
A line segment is part of a line that connects two points without extending beyond them.
In our case, the line segment connects the points \((-2, 1)\) and \(\frac{3}{5}, -4\).
Finding the midpoint means finding a point that divides this segment into two equal halves.
Using the midpoint formula \(\text{Midpoint} = \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\) helps us accomplish this.
By plugging in the given coordinates and simplifying accordingly, we determine the exact middle point of the line segment, \(\frac{-7}{10}, \frac{-3}{2}\).

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Most popular questions from this chapter

Minimum Average cost The average cost per hour in dollars, \(\bar{C},\) of producing \(x\) riding lawn mowers can be modeled by the function $$ \bar{C}(x)=0.3 x^{2}+21 x-251+\frac{2500}{x} $$ (a) Use a graphing utility to graph \(\bar{C}=\bar{C}(x)\) (b) Determine the number of riding lawn mowers to produce in order to minimize average cost. (c) What is the minimum average cost?

The relationship between the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\) scales for measuring temperature is given by the equation $$ F=\frac{9}{5} C+32 $$ The relationship between the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and \(\mathrm{Kelvin}(\mathrm{K})\) scales is \(K=C+273 .\) Graph the equation \(F=\frac{9}{5} C+32\) using degrees Fahrenheit on the \(y\) -axis and degrees Celsius on the \(x\) -axis. Use the techniques introduced in this section to obtain the graph showing the relationship between Kelvin and Fahrenheit temperatures.

(a) Graph \(f(x)=|x-3|-3\) using transformations. (b) Find the area of the region that is bounded by \(f\) and the \(x\) -axis and lies below the \(x\) -axis.

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=-x^{2}+3 x-2\)

\(h(x)=-2 x^{2}+x\) (a) Find the average rate of change from 0 to 3 . (b) Find an equation of the secant line containing \((0, h(0))\) and \((3, h(3))\)
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