/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 75 A line segment has \(\left(x_{1}... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 75

A line segment has \(\left(x_{1}, y_{1}\right)\) as one endpoint and \(\left(x_{m}, y_{m}\right)\) as its midpoint. Find the other endpoint \(\left(x_{2}, y_{2}\right)\) of the line segment in terms of \(x_{1}, y_{1}, x_{m},\) and \(y_{m}\).

Short Answer

Expert verified
\( (x_{2}, y_{2}) = (2x_{m} - x_{1}, 2y_{m} - y_{1}) \)

Step by step solution

01

Recall the Midpoint Formula

The first step involves recalling the midpoint formula. If \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) are the end points of a line segment, its midpoint \((x_{m}, y_{m})\) is given by \((x_{m}, y_{m}) = \left(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}\right)\.
02

Plug in Given Values and Solve for Unknowns

In this problem, we are given the midpoint \((x_{m}, y_{m})\) and one endpoint \((x_{1}, y_{1})\), and we are asked to find the other endpoint \((x_{2}, y_{2})\). Plugging the given values into the midpoint formula, we get \(x_{m} = \frac{x_{1} + x_{2}}{2}\) and \(y_{m} = \frac{y_{1} + y_{2}}{2}\). Solving these equations for \(x_{2}\) and \(y_{2}\), we obtain \(x_{2} = 2x_{m} - x_{1}\) and \(y_{2} = 2y_{m} - y_{1}\).
03

Write Down the Solution

After performing the calculations in step 2, we find that the coordinates of the other endpoint of the line segment are \((x_{2}, y_{2}) = (2x_{m} - x_{1}, 2y_{m} - y_{1})\).

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Most popular questions from this chapter

Use the given value of \(k\) to complete the table for the inverse variation model $$y=\frac{k}{x^{2}}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=\frac{k}{x^{2}} & & & & & \\ \hline \end{array}$$ $$ k=5 $$

Determine whether the variation model is of the form \(y=k x\) or \(y=k / x,\) and find \(k .\) Then write \(a\) model that relates \(y\) and \(x\). $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 24 & 12 & 8 & 6 & \frac{24}{5} \\ \hline \end{array} $$

Find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) \(y\) is inversely proportional to \(x .(y=7\) when \(x=4\).)

On a yardstick with scales in inches and centimeters, you notice that 13 inches is approximately the same length as 33 centimeters. Use this information to find a mathematical model that relates centimeters \(y\) to inches \(x\). Then use the model to find the numbers of centimeters in 10 inches and 20 inches.

The winning times (in minutes) in the women's 400-meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. $$\begin{array}{lll} (1948,5.30) & (1972,4.32) & (1996,4.12) \\ (1952,5.20) & (1976,4.16) & (2000,4.10) \\ (1956,4.91) & (1980,4.15) & (2004,4.09) \\ (1960,4.84) & (1984,4.12) & (2008,4.05) \\ (1964,4.72) & (1988,4.06) & \\ (1968,4.53) & (1992,4.12) & \end{array}$$ A linear model that approximates the data is \(y=-0.020 t+5.00,\) where \(y\) represents the winning time (in minutes) and \(t=0\) represents \(1950 .\) Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: International Olympic Committee)
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