Problem 69 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation for the line that bisects the line segment from \((-2,3)\) to \((1,-2)\) and is at right angles to this line segment.

Step-by-Step Solution

Answer

The equation is \( y = \frac{3}{5}x + \frac{4}{5}.\)

Step 1: Find Midpoint of the Line Segment

The midpoint of a line segment with endpoints \((-2, 3)\) and \(1, -2)\) is calculated using the midpoint formula, given by: \( M(x, y) = \left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}\right) \).Substitute the given points into the formula: \( M = \left(\frac{-2 + 1}{2}, \frac{3 - 2}{2}\right) = \left(\frac{-1}{2}, \frac{1}{2}\right).\)So, the midpoint is \( \left(\frac{-1}{2}, \frac{1}{2}\right).\)

Step 2: Calculate the Slope of the Given Line Segment

The slope \( m \) of the line segment through points \((-2, 3)\) and \(1, -2)\) is found using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).Substitute the given points: \( m = \frac{-2 - 3}{1 + 2} = \frac{-5}{3}.\)Thus, the slope of the line segment is \(-\frac{5}{3}.\)

Step 3: Determine the Negative Reciprocal Slope

To find the slope of a line perpendicular to the given line segment, take the negative reciprocal of the original slope.For a slope of \(-\frac{5}{3}\), the negative reciprocal is \(\frac{3}{5}\).Thus, the slope of the perpendicular bisector is \(\frac{3}{5}.\)

Step 4: Write the Equation of the Perpendicular Bisector

Now you have the slope \(\frac{3}{5}\) and the midpoint \(\left(\frac{-1}{2}, \frac{1}{2}\right)\). Use the point-slope form of a line equation, given by \( y - y_1 = m(x - x_1) \).Plug in the midpoint and the perpendicular slope: \( y - \frac{1}{2} = \frac{3}{5}\left(x + \frac{1}{2}\right).\)Simplify this equation: Start by distributing the slope: \( y - \frac{1}{2} = \frac{3}{5}x + \frac{3}{10}.\)Add \(\frac{1}{2}\) to both sides: \( y = \frac{3}{5}x + \frac{3}{10} + \frac{5}{10} = \frac{3}{5}x + \frac{8}{10}.\)The equation can be simplified to \( y = \frac{3}{5}x + \frac{4}{5}.\)Thus, the equation of the line that bisects the given line segment at a right angle is \( y = \frac{3}{5}x + \frac{4}{5}.\)

Key Concepts

Midpoint Formula

To find the midpoint of a line segment with given endpoints, you use the midpoint formula. This is a simple way to average the x-coordinates and the y-coordinates of the endpoints separately. The formula is: \[ M(x, y) = \left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}\right) \] Where

• \((x_1, y_1)\) and \((x_2, y_2)\) are the given endpoints of the line segment.
• \(M(x,y)\) represents the midpoint coordinates.

This formula helps find the point that is exactly halfway between the two endpoints. For example, in the problem, our two points are \((-2, 3)\) and \((1, -2)\). Substituting these values into our formula gives us the midpoint \(\left(\frac{-1}{2}, \frac{1}{2}\right)\). This midpoint is crucial when you're solving problems such as finding a perpendicular bisector, as it is a point the bisector passes through.

Slope of a Line

The slope of a line measures its steepness and is calculated using a very straightforward formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Where

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of any two different points on the line.
• \(m\) is the slope of the line.

In the original exercise, the endpoints of the line segment were \((-2, 3)\) and \((1, -2)\). Plugging these into our formula gives a slope of \(-\frac{5}{3}\). The slope tells us that for every three units moved horizontally, the line moves five units vertically in the opposite direction due to the negative sign. Understanding slope is fundamental for analyzing the direction and angle of a line, which is important in determining perpendicular relationships.

Point-Slope Form

Once we determine a line's slope and know a point on the line, we use the point-slope form to write its equation. This handy formula is written as: \[ y - y_1 = m(x - x_1) \] Where

• \((x_1, y_1)\) is a known point on the line.
• \(m\) is the slope of the line.

In the context of our problem, the known point is the midpoint \( \left( \frac{-1}{2}, \frac{1}{2} \right) \), and the slope is the negative reciprocal of the original line, which is \( \frac{3}{5} \). Inserting these into the formula provides the original equation of the perpendicular bisector: \[ y - \frac{1}{2} = \frac{3}{5}(x + \frac{1}{2}) \]. This form is particularly useful in problems where you need to quickly establish the relationship between a given point and a line.

Negative Reciprocal

Understanding the concept of negative reciprocals is essential when dealing with perpendicular lines. The slopes of two lines that are perpendicular to each other are negative reciprocals. This means you swap the numerator and denominator of the original slope and add a negative sign (or flip its sign). For example, if you have a slope of \(-\frac{5}{3}\), the negative reciprocal would be \(\frac{3}{5}\). To find this:

• Swap the numbers \(5\) and \(3\) to turn the fraction into \(\frac{3}{5}\).
• Change the sign from negative to positive.

This new slope, \(\frac{3}{5}\), indicates the slope of a line perpendicular to the original line. Recognizing and using negative reciprocals is key in geometric problems involving perpendicular lines, such as finding the perpendicular bisector of a line segment.