Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, giving it in slope-intercept form if possible. 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 \(y = -\frac{7}{4}x + \frac{63}{8}\) in slope-intercept form.

Step 1: Calculate the Midpoint

To find the midpoint of the line segment connecting the points \((-3,5)\) and \((4,9)\), we use the midpoint formula:\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Substitute in the values:\[\left( \frac{-3 + 4}{2}, \frac{5 + 9}{2} \right) = \left( \frac{1}{2}, 7 \right)\]So, the midpoint is \(\left( \frac{1}{2}, 7 \right)\).

Step 2: Find the Slope of the Given Line Segment

Using the slope formula\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Calculate the slope of the line segment connecting \((-3,5)\) and \((4,9)\):\[m = \frac{9 - 5}{4 - (-3)} = \frac{4}{7}\]The slope of the line segment is \(\frac{4}{7}\).

Step 3: Determine the Slope of the Perpendicular Bisector

The slope of the line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular bisector is:\[m_{pb} = -\frac{1}{\left(\frac{4}{7}\right)} = -\frac{7}{4}\]

Step 4: Write the Equation of the Perpendicular Bisector in Point-Slope Form

Using the point-slope form of an equation, which is \[y - y_1 = m(x - x_1)\]Substitute the midpoint \(\left( \frac{1}{2}, 7 \right)\) and the slope \(-\frac{7}{4}\):\[y - 7 = -\frac{7}{4}\left(x - \frac{1}{2}\right)\]This is the equation in point-slope form.

Step 5: Convert the Equation to Slope-Intercept Form

Begin by distributing the slope in the point-slope equation:\[y - 7 = -\frac{7}{4}x + \frac{7}{8}\]Add 7 to both sides to solve for \(y\):\[y = -\frac{7}{4}x + \frac{7}{8} + 7\]Convert 7 to \(\frac{56}{8}\) to combine like terms:\[y = -\frac{7}{4}x + \frac{63}{8}\]The slope-intercept form of the line is \(y = -\frac{7}{4}x + \frac{63}{8}\).

Key Concepts

Midpoint Formula

Finding the midpoint of a line segment is an essential step in various geometrical calculations. The midpoint is the point that is exactly halfway between the two endpoints of a line segment. To find it, you use the midpoint formula:

• Take the average of the x-coordinates of the two points.
• Take the average of the y-coordinates of the two points.

The formula is: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]For example, if you have points \((-3,5)\) and \((4,9)\), applying the formula gives:\[\left( \frac{-3 + 4}{2}, \frac{5 + 9}{2} \right) = \left( \frac{1}{2}, 7 \right)\]This result tells us that the midpoint between the two points is \( \left( \frac{1}{2}, 7 \right) \). By understanding this, students will have a starting point for constructing perpendicular bisectors or dividing line segments.

Slope Formula

The slope is a measure of the steepness or the incline of a line. It's essential for understanding how lines relate to each other on a graph. The slope formula calculates this steepness between two points, and is expressed as: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]This means you subtract the y-coordinates and the x-coordinates respectively, and then divide the differences. In our example, connecting points \((-3,5)\) and \((4,9)\), the slope is: \[m = \frac{9 - 5}{4 - (-3)} = \frac{4}{7}\]This tells us the line rises 4 units for every 7 units it runs horizontally. Grasping this concept is crucial, especially when looking at parallel and perpendicular lines in geometry.

Slope-Intercept Form

The slope-intercept form is one of the most commonly used forms of a linear equation. It allows for easy graphing and gives a clear view of how a line behaves by using the format: \[ y = mx + b \]Here, \(m\) represents the slope, and \(b\) is the y-intercept, where the line crosses the y-axis. By converting equations into this form, you can quickly see these characteristics. For example, once the equation of our perpendicular bisector is simplified into this style, it is:\[ y = -\frac{7}{4}x + \frac{63}{8} \]This immediately tells you that the line descends due to the negative slope and crosses the y-axis at the value \( \frac{63}{8} \). It's a straightforward way to interpret and visualize linear relationships.

Point-Slope Form

The point-slope form of a linear equation is particularly useful when you know a point on the line and the slope. It is written as: \[ y - y_1 = m(x - x_1) \]Here, \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope. This form is advantageous in creating an equation when these specific pieces of information are available. In our example of finding the perpendicular bisector, by using the midpoint \( \left( \frac{1}{2}, 7 \right) \) and the slope \(-\frac{7}{4}\), we write: \[ y - 7 = -\frac{7}{4}\left(x - \frac{1}{2}\right) \]This equation clearly represents a line without needing the y-intercept right away, making it a powerful tool when dealing with immediate calculations from known data.