Problem 58 - A Graphical Approach to College Algebra

Question

Find the equation of the line that is the perpendicular bisector of the line segment connecting \((-3,5)\) and \((4,9)\)

Step-by-Step Solution

Answer

The equation is \(y = -\frac{7}{4}x + \frac{63}{8}\).

Step 1: Find the Midpoint of the Line Segment

To find the equation of the perpendicular bisector, we first find the midpoint of the line segment connecting the two points \((-3, 5)\) and \(4, 9)\). Use the midpoint formula:\[\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Substitute \(x_1 = -3, y_1 = 5, x_2 = 4, y_2 = 9\):\[\text{Midpoint} = \left( \frac{-3 + 4}{2}, \frac{5 + 9}{2} \right) = \left( \frac{1}{2}, 7 \right)\]The midpoint is \(\left( \frac{1}{2}, 7 \right)\).

Step 2: Calculate the Slope of the Original Line

Next, find the slope of the line segment connecting the points \((-3, 5)\) and \((4, 9)\). Use the slope formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute \(y_2 = 9, y_1 = 5, x_2 = 4, x_1 = -3\):\[m = \frac{9 - 5}{4 + 3} = \frac{4}{7}\]The slope of the original line is \(\frac{4}{7}\).

Step 3: Determine the Slope of the Perpendicular Bisector

The perpendicular bisector will have a slope that is the negative reciprocal of the slope of the line segment. The negative reciprocal of \(\frac{4}{7}\) is \(-\frac{7}{4}\). Hence, the slope of the perpendicular bisector is \(-\frac{7}{4}\).

Step 4: Write the Equation of the Perpendicular Bisector

Use the point-slope form of a line equation with the midpoint \(\left( \frac{1}{2}, 7 \right)\) and the slope \(-\frac{7}{4}\):\[y - y_1 = m(x - x_1)\]Substitute \(m = -\frac{7}{4}, x_1 = \frac{1}{2}, y_1 = 7\):\[y - 7 = -\frac{7}{4}\left(x - \frac{1}{2}\right)\]Distribute and simplify:\[y - 7 = -\frac{7}{4}x + \frac{7}{8}\]Adding 7 to both sides gives:\[y = -\frac{7}{4}x + \frac{7}{8} + \frac{56}{8}\]\[y = -\frac{7}{4}x + \frac{63}{8}\]So, the equation of the perpendicular bisector is \(y = -\frac{7}{4}x + \frac{63}{8}\).

Key Concepts

Understanding the Midpoint Formula

The midpoint formula is an essential tool in coordinate geometry used for finding the center point between two given points in a plane. When you have a line segment connecting two points, say \((-3,5)\) and \(4,9)\), the midpoint formula helps you calculate the halfway point. This formula is:

• \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

To determine the midpoint, substitute the given coordinates into the formula:

• \( x_1 = -3, y_1 = 5, x_2 = 4, y_2 = 9 \)

The calculation follows:

• \[ \text{Midpoint} = \left( \frac{-3 + 4}{2}, \frac{5 + 9}{2} \right) = \left( \frac{1}{2}, 7 \right) \]

This midpoint, \(\frac{1}{2}, 7\), is an essential reference point when determining the equation of a line, particularly for perpendicular bisectors.

Calculating the Slope of a Line

Understanding slope calculation is crucial in determining how steep a line is on a graph. The slope, often denoted by \(m\), is calculated using the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Essentially, this formula divides the change in the vertical direction (\(y\)) by the change in the horizontal direction (\(x\)) between two points.For our points \((-3, 5)\) and \(4, 9)\), substitute them into the formula:

• \( y_2 = 9, y_1 = 5, x_2 = 4, x_1 = -3 \)

The slope is calculated as:

• \[ m = \frac{9 - 5}{4 + 3} = \frac{4}{7} \]

This result tells us that for every 7 units the line moves horizontally, it moves 4 units vertically. When looking for a perpendicular bisector, the critical step is finding the negative reciprocal of this slope, which flips and changes the sign.

Application of the Point-Slope Form

The point-slope form is a fundamental equation in algebra useful for defining lines with a known slope \(m\) passing through a specific point. It is expressed as:

• \[ y - y_1 = m(x - x_1) \]

Where \(x_1, y_1\) is a point on the line and \(m\) is the slope. When you use this form, you can easily derive the equation of a line.For the perpendicular bisector of the points \((-3, 5)\) and \(4, 9)\), the slope needed is the negative reciprocal of the original line's slope \(\frac{4}{7}\), resulting in \(-\frac{7}{4}\). Using the midpoint \(\frac{1}{2}, 7\) as your reference point, substitute into the point-slope form:

• \[ y - 7 = -\frac{7}{4}(x - \frac{1}{2}) \]

Expand and simplify:

• \[ y - 7 = -\frac{7}{4}x + \frac{7}{8} \]
• Add 7 to both sides: \[ y = -\frac{7}{4}x + \frac{7}{8} + \frac{56}{8} \]
• Final equation: \[ y = -\frac{7}{4}x + \frac{63}{8} \]

This equation now represents the perpendicular bisector of the original line segment.