Problem 57 - A Graphical Approach to College Algebra

Question

Find the equation of the line that is the perpendicular bisector of the line segment connecting \((-4,2)\) and \((2,10)\)

Step-by-Step Solution

Answer

The equation of the perpendicular bisector is \(y = -\frac{3}{4}x + \frac{21}{4}\).

Step 1: Find the Midpoint

To find the midpoint of the segment connecting \((-4, 2)\) and \((2, 10)\), use the formula for the midpoint of a line segment: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Substituting in the given points, we have \(\left( \frac{-4 + 2}{2}, \frac{2 + 10}{2} \right)\) which simplifies to the midpoint \((-1, 6)\).

Step 2: Calculate the Original Slope

The slope of the line joining the points \((-4, 2)\) and \((2, 10)\) can be calculated using the formula for slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points into the formula gives us \(m = \frac{10 - 2}{2 + 4} = \frac{8}{6} = \frac{4}{3}\).

Step 3: Find the Perpendicular Slope

The slope of the perpendicular bisector is the negative reciprocal of the slope of the original line. Thus, if the slope of the original line is \(\frac{4}{3}\), the slope of the perpendicular bisector is \(-\frac{3}{4}\).

Step 4: Write the Equation of the Perpendicular Bisector

Use the point-slope form of a line equation: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope found in Step 3 and \((x_1, y_1)\) is the midpoint from Step 1. Substituting \(-\frac{3}{4}\) for \(m\) and \((-1, 6)\) for \((x_1, y_1)\), the equation becomes \(y - 6 = -\frac{3}{4}(x + 1)\). Simplifying gives the equation of the perpendicular bisector as \(y = -\frac{3}{4}x + \frac{21}{4}\).

Key Concepts

Midpoint

Understanding the concept of a midpoint is essential when working with line segments. The midpoint is essentially the average point between two endpoints of a line segment. This point divides the segment into two equal halves.
To find the midpoint, use the formula:

• Midpoint coordinates: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right).\)

For example, if you have endpoints at \((-4, 2)\) and \((2, 10)\), substitute the coordinates into the formula:

• \(\left( \frac{-4 + 2}{2}, \frac{2 + 10}{2} \right)\)
• This simplifies to the midpoint: \((-1, 6)\).

This midpoint is used as a reference point when writing the equation of the perpendicular bisector.

Slope Formula

The slope of a line is a measure of its steepness and is represented by the letter \(m\). The slope is crucial for understanding how two lines are related, especially when determining perpendicular lines.
To find the slope between two points, use the slope formula:

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

For instance, with points \((-4, 2)\) and \((2, 10)\), apply these values:

• \(m = \frac{10 - 2}{2 + 4} = \frac{8}{6} = \frac{4}{3}\)

This original slope helps to find the slope of a perpendicular line, which will be its negative reciprocal. In this case, the perpendicular slope is \(-\frac{3}{4}\). This is critical for constructing a perpendicular bisector as it shows the direction the line needs to incline.

Point-Slope Form

Writing the equation of a line is straightforward with the point-slope form, making it an excellent choice when you have a point and slope.
The point-slope form equation is as follows:

• \(y - y_1 = m(x - x_1)\)

Here, \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope.
For a perpendicular bisector, use the midpoint as \((x_1, y_1)\) and the perpendicular slope found earlier. For example, with midpoint \((-1, 6)\) and slope \(-\frac{3}{4}\):

• \(y - 6 = -\frac{3}{4}(x + 1)\)
• Simplifying this, we arrive at the line equation: \(y = -\frac{3}{4}x + \frac{21}{4}\).

This form allows easy manipulation to the slope-intercept form, enhancing understanding of where the line crosses the coordinate axes.