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 perpendicular bisector has the equation: \( y = \frac{3}{5}x + \frac{4}{5} \).

Step 1: Find the Midpoint

To find the midpoint of the line segment connecting the points \( (-2,3) \) and \( (1,-2) \), use the midpoint formula \((x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Substitute the given values:\[ x_m = \frac{-2 + 1}{2} = \frac{-1}{2}, \quad y_m = \frac{3 + (-2)}{2} = \frac{1}{2} \]So the midpoint is \( \left( \frac{-1}{2}, \frac{1}{2} \right) \).

Step 2: Find the Slope of the Line Segment

The slope (m) of the line segment can be calculated using the formula \((y_2 - y_1) / (x_2 - x_1)\). Substitute the given values:\[ m = \frac{-2 - 3}{1 + 2} = \frac{-5}{3} \]

Step 3: Find the Slope of the Perpendicular Bisector

Perpendicular lines have slopes that are negative reciprocals of each other. So, the slope of the perpendicular bisector is the negative reciprocal of \(-\frac{5}{3}\), which is \frac{3}{5}\.

Step 4: Write the Equation of the Perpendicular Bisector

Now use the point-slope form of the equation of a line, \(y - y_m = m(x - x_m)\), to write the equation of the line:Substitute the values of the midpoint \( (\frac{-1}{2}, \frac{1}{2}) \) and the slope \( \frac{3}{5} \):\[ y - \frac{1}{2} = \frac{3}{5}(x + \frac{1}{2}) \]Convert this into the slope-intercept form (y = mx + b):\[ y = \frac{3}{5}x + \frac{3}{10} + \frac{1}{2} \]\[ y = \frac{3}{5}x + \frac{3}{10} + \frac{5}{10} \]\[ y = \frac{3}{5}x + \frac{8}{10} \]Simplify:\[ y = \frac{3}{5}x + \frac{4}{5} \]

Key Concepts

Midpoint Formula

The midpoint formula is a simple tool used to find the middle point between two endpoints on a line segment. If you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the formula to find the midpoint is given by:

• \( x_m = \frac{x_1 + x_2}{2} \)
• \( y_m = \frac{y_1 + y_2}{2} \)

For instance, in our original exercise, to find the midpoint of the segment connecting \( (-2, 3) \) and \( (1, -2) \), substitute the coordinates into the formula:

• \( x_m = \frac{-2 + 1}{2} = \frac{-1}{2} \)
• \( y_m = \frac{3 - 2}{2} = \frac{1}{2} \)

Therefore, the midpoint is the point \( \left(\frac{-1}{2}, \frac{1}{2}\right) \). This midpoint represents the exact center of the line segment and is crucial for finding a perpendicular bisector.

Slope of a Line

The slope of a line measures the steepness or the tilt of the line, and it is an important concept in understanding linear equations. Calculated as the change in y divided by the change in x between two points, it is expressed with the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In the original problem, the slope of the line connecting \( (-2, 3) \) and \( (1, -2) \) is found by:

• \( m = \frac{-2 - 3}{1 + 2} = \frac{-5}{3} \)

This negative slope indicates that the line runs downwards from left to right. To find the slope of a line \ that is perpendicular to another, you use the negative reciprocal of the original slope. So, the slope of the perpendicular bisector is \( \frac{3}{5} \), which represents the tilt of the bisector line that is the subject of our exercise.

Point-Slope Form

The point-slope form of a linear equation is extremely useful when you know a point on the line and the slope but not necessarily the y-intercept. The formula is given by:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. It allows you to quickly write the equation of a line. In our solution, we have:

• \( y - \frac{1}{2} = \frac{3}{5}(x + \frac{1}{2}) \)

This equation comes from using the midpoint \( \left(\frac{-1}{2}, \frac{1}{2}\right) \) as our point on the line, with \( \frac{3}{5} \) as the slope. Using this form helps transition easily into other forms, like the slope-intercept form.

Slope-Intercept Form

The slope-intercept form is one of the most popular forms for the equation of a line. It's written as:

• \( y = mx + b \)

Here, \( m \) is the slope, and \( b \) is the y-intercept, where the line crosses the y-axis. This form is handy for quickly identifying both the slope and y-intercept, which provide valuable insights about the line's direction and position. In our example, using the transformations from the point-slope form:

• \( y = \frac{3}{5}x + \frac{4}{5} \)

This final equation clearly shows the slope as \( \frac{3}{5} \) and y-intercept as \( \frac{4}{5} \), making it clear where the perpendicular bisector intersects the y-axis and its overall direction.