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Chapter 8: Problem 92

Find an equation of the perpendicular bisector of the line segment whose endpoints are given. See the previous example. $$ (5,8) ;(7,2) $$

Short Answer

Expert verified
The equation of the perpendicular bisector is: \(y = \frac{1}{3}x + 3\).

Step by step solution

01

Find the Midpoint

The midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). For the given points \((5,8)\) and \((7,2)\), the midpoint is:\(\left( \frac{5 + 7}{2}, \frac{8 + 2}{2} \right) = (6, 5)\).. This is the point where the perpendicular bisector will pass through.
02

Calculate the Slope of the Original Line

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).For the points \((5,8)\) and \((7,2)\), substitute the values to get: \( m = \frac{2 - 8}{7 - 5} = -3 \).This means the slope of the line joining these two points is \(-3\).
03

Determine the Slope of the Perpendicular Bisector

The slope of a perpendicular bisector is the negative reciprocal of the slope of the original line.Since the slope of the original line is \(-3\), the slope of the perpendicular bisector is \(\frac{1}{3}\).
04

Write the Equation of the Perpendicular Bisector

Using the point-slope form of a line \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line, substitute \(m = \frac{1}{3}\) and the midpoint \((6, 5)\):\(y - 5 = \frac{1}{3}(x - 6)\).Expanding this equation gives:\(y - 5 = \frac{1}{3}x - 2\)which simplifies to\(y = \frac{1}{3}x + 3\).This is the equation of the perpendicular bisector.

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Key Concepts

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

Midpoint Formula
The midpoint formula is essential for finding the center location of a line segment. This center point is where the perpendicular bisector will intersect the segment. To find the midpoint, use the endpoints' coordinates. The formula is: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Applying this to the given points \((5, 8)\) and \((7, 2)\), we compute: * The average of the x-coordinates: \( \frac{5 + 7}{2} = 6 \). * The average of the y-coordinates: \( \frac{8 + 2}{2} = 5 \). Thus, the midpoint is \((6, 5)\), which will be the exact point the perpendicular bisector passes through.
Slope Calculation
Understanding how to calculate the slope of a line is crucial because it helps determine the angle or steepness of the line. The slope formula is: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The slope \(m\) indicates the ratio of vertical change to horizontal change between two points. For the line segment's endpoints \((5,8)\) and \((7,2)\): * Change in y: \(2 - 8 = -6\) * Change in x: \(7 - 5 = 2\) Therefore, the slope \(m\) is \(-3\). This slope is critical as its negative reciprocal will give us the slope of the perpendicular bisector.
Point-Slope Form
The point-slope form allows you to write the equation of a line if you know a point on the line and the slope. The formula is: \( y - y_1 = m(x - x_1) \), where: * \(m\) is the slope * \((x_1, y_1)\) is a point on the line For our perpendicular bisector: * We have the midpoint \((6, 5)\) * The slope \(m\) is \(\frac{1}{3}\) (negative reciprocal of \(-3\)) Inserting these into the point-slope formula gives us: \( y - 5 = \frac{1}{3}(x - 6) \). This equation provides a straightforward method to visualize the line's behavior.
Line Equations
Line equations like the one we derived for the perpendicular bisector offer a way to describe every point on the line within a coordinate plane. After using the point-slope form, we can expand to the slope-intercept form, \( y = mx + b \), which provides intuitive insight into the line's slope and intercept. From \( y - 5 = \frac{1}{3}(x - 6) \), expanding and simplifying yields: * Distribute \(\frac{1}{3}\): \( y - 5 = \frac{1}{3}x - 2 \) * Solve for \(y\): \( y = \frac{1}{3}x + 3 \) This final equation is in slope-intercept form, showing the line crosses the y-axis at \(3\). Such equations are invaluable for graphing and understanding geometric relationships.

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Most popular questions from this chapter

Del Monte Fruit Company recently released a new applesauce. By the end of its first year, profits on this product amounted to \(\$ 30.000 .\) The anticipated profit for the end of the fourth year is \(\$ 66,000\). The ratio of change in time to change in profit is constant. Let \(x\) be years and \(P\) be profit. a. Write a linear function \(P(x)\) that expresses profit as a function of time. b. Use this function to predict the company's profit at the end of the seventh year. c. Predict when the profit should reach \(\$ 126,000\).

Complete the following table for the inverse variation \(y=\frac{k}{x}\) over each given value of \(k .\) Plot the points on a rectangular coordinate system. $$ \begin{array}{c|c|c|c|c|c|} \hline x & {\frac{1}{4}} & {\frac{1}{2}} & {1} & {2} & {4} \\ \hline y {=\frac{k}{x}} & {} & {} & {} & {} \\ \hline \end{array} $$ $$ k=1 $$

Find the equation of each line. Write the equation using standard notation unless indicated otherwise. Through \((3,5)\); perpendicular to the line \(2 x-y=8\)

Simplify. $$ -|7| $$

Complete the following table for the inverse variation \(y=\frac{k}{x}\) over each given value of \(k .\) Plot the points on a rectangular coordinate system. $$ \begin{array}{c|c|c|c|c|c|} \hline x & {\frac{1}{4}} & {\frac{1}{2}} & {1} & {2} & {4} \\ \hline y {=\frac{k}{x}} & {} & {} & {} & {} \\ \hline \end{array} $$ $$ k=3 $$
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