Problem 56 - Contemporary Precalculus

Question

Find the equation of the perpendicular bisector of the line segment joining the two given points. $$(-6,2),(6,-7)$$

Step-by-Step Solution

Answer

Answer: The equation of the perpendicular bisector is 8x - 6y = 15.

Step 1: Find the midpoint of the line segment joining the given points

To find the midpoint \((x_m, y_m)\) of the line segment joining the points \((-6, 2)\) and \((6, -7)\), we can use the midpoint formula: $$x_m = \frac{x_1+x_2}{2}$$ $$y_m = \frac{y_1+y_2}{2}$$ So, in our case, \(x_m = \frac{-6+6}{2}\) and \(y_m = \frac{2-7}{2}\).

Step 2: Calculate the midpoint coordinates

Computing the midpoint coordinates, we get: $$x_m = \frac{-6+6}{2}=0$$ $$y_m = \frac{2-7}{2}=-\frac{5}{2}$$ Thus, the midpoint of the line segment is \((0, -\frac{5}{2})\).

Step 3: Determine the slope of the original line

To find the slope \(m\) of the line containing points \((-6, 2)\) and \((6, -7)\), we can use the slope formula: $$m = \frac{y_2-y_1}{x_2-x_1}$$ Substituting the given coordinates, we get: $$m = \frac{-7-2}{6-(-6)}$$

Step 4: Calculate the slope of the original line

Computing the slope, we get: $$m = \frac{-9}{12} = -\frac{3}{4}$$ So, the slope of the original line is \(-\frac{3}{4}\).

Step 5: Find the slope of a perpendicular line

The slope of a line perpendicular to the original line will be the negative reciprocal of the original slope. So in this case: $$m_{\perp} = \frac{4}{3}$$ Hence, the slope of the line perpendicular to the original line is \(\frac{4}{3}\).

Step 6: Use point-slope form to write the equation of the perpendicular bisector

Using the point-slope form of a linear equation, which is: $$y - y_m = m_{\perp}(x - x_m)$$ Substitute the values for \(x_m\), \(y_m\) and \(m_{\perp}\) from steps 2 and 5 and we get: $$y - \left(-\frac{5}{2}\right) = \frac{4}{3}(x - 0)$$

Step 7: Simplify the equation to get the final result

Simplifying the equation, we have: $$y + \frac{5}{2} = \frac{4}{3}x$$ Multiplying both sides by \(6\) to eliminate fractions, we get: $$6y + 15 = 8x$$ Rearrange the terms to obtain the equation of the perpendicular bisector: $$8x - 6y = 15$$

Key Concepts

Midpoint

The midpoint of a line segment is a point that lies exactly halfway between two given endpoints. To find the midpoint, use the midpoint formula:

• For the x-coordinate: \( x_m = \frac{x_1+x_2}{2} \)
• For the y-coordinate: \( y_m = \frac{y_1+y_2}{2} \)

Apply this formula to the points \((-6, 2)\) and \((6, -7)\):

• \(x_m = \frac{-6 + 6}{2} = 0\)
• \(y_m = \frac{2 - 7}{2} = -\frac{5}{2}\)

Thus, the midpoint is \((0, -\frac{5}{2})\), which divides the line segment into two equal parts.

Slope of a Line

The slope of a line measures its steepness or incline and is crucial for understanding the line's direction. Slope \( m \) is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In this exercise, using points \((-6, 2)\) and \((6, -7)\), substitute the values:

• \( m = \frac{-7 - 2}{6 - (-6)}\)
• \( m = \frac{-9}{12} = -\frac{3}{4} \)

The slope \(-\frac{3}{4}\) tells us that the line moves 3 units downward for every 4 units it moves to the right.

Point-Slope Form

The point-slope form of a line's equation is very handy for quickly writing the equation of a line when you know a point on the line and its slope. The formula is:\[ y - y_1 = m(x - x_1) \]Where \( m \) is the slope, and \((x_1, y_1)\) is the known point. For the perpendicular bisector, our point is the midpoint \((0, -\frac{5}{2})\), and the perpendicular slope is \( \frac{4}{3} \). Plug these values in:

• \( y + \frac{5}{2} = \frac{4}{3}(x - 0) \)

This sets up the equation in a form that can easily be simplified further.

Negative Reciprocal

A negative reciprocal is crucial when finding the slope of a line perpendicular to another. If you have the slope of a line, its perpendicular slope is the negative reciprocal of the original slope.

• Start with the original slope \(-\frac{3}{4}\).
• Flip the fraction to get \(\frac{4}{3}\).
• Change the sign to get \(\frac{4}{3}\).

The perpendicular slope ensures the lines intersect at a right angle. This property is essential for finding the perpendicular bisector's equation, as demonstrated when we used \(\frac{4}{3}\) as the slope in the point-slope form of the line equation.