Problem 77 - Algebra for College Students

Question

The problem of finding the perpendicular bisector of a line segment presents itself often in the study of analytic geometry. As with any problem of writing the equation of a line, you must determine the slope of the line and a point that the line passes through. A perpendicular bisector passes through the midpoint of the line segment and has a slope that is the negative reciprocal of the slope of the line segment. The problem can be solved as follows: Find the perpendicular bisector of the line segment between the points \((1,-2)\) and \((7,8)\). The midpoint of the line segment is \(\left(\frac{1+7}{2}, \frac{-2+8}{2}\right)\) \(=(4,3)\). \(=(4,3)\). The slope of the line segment is \(m=\frac{8-(-2)}{7-1}\) \(=\frac{10}{6}=\frac{5}{3}\) Hence the perpendicular bisector will pass through the point \((4,3)\) and have a slope of \(m=-\frac{3}{5}\). $$ \begin{aligned} y-3 &=-\frac{3}{5}(x-4) \\ 5(y-3) &=-3(x-4) \\ 5 y-15 &=-3 x+12 \\ 3 x+5 y &=27 \end{aligned} $$ Thus the equation of the perpendicular bisector of the line segment between the points \((1,-2)\) and \((7,8)\) is \(3 x+5 y=27 .\) Find the perpendicular bisector of the line segment between the points for the following. Write the equation in standard form. (a) \((-1,2)\) and \((3,0)\) (b) \((6,-10)\) and \((-4,2)\) (c) \((-7,-3)\) and \((5,9)\) (d) \((0,4)\) and \((12,-4)\)

Step-by-Step Solution

Answer

(a) \(2x - y = 1\); (b) \(5x - 6y = 29\); (c) \(x + y = 2\); (d) \(3x - 2y = 18\).

Step 1: Calculate Midpoint (a)

To find the midpoint for part (a) with points \((-1,2)\) and \((3,0)\), use the midpoint formula: \( \left(\frac{-1+3}{2}, \frac{2+0}{2}\right) \). This simplifies to \((1, 1)\).

Step 2: Calculate Slope (a)

Calculate the slope of the line segment for part (a) using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0-2}{3+1} = -\frac{1}{2} \).

Step 3: Find Negative Reciprocal Slope (a)

The slope of the perpendicular bisector is the negative reciprocal of the line's slope. For part (a), this becomes \( m = 2 \).

Step 4: Equation of Perpendicular Bisector (a)

Using point-slope form \(y - y_1 = m(x - x_1)\), substitute the midpoint \((1, 1)\) and the perpendicular slope \(2\):\[y - 1 = 2(x - 1)\]. Expand and simplify to standard form: \[2x - y = 1\].

Step 5: Calculate Midpoint (b)

For part (b), points \((6,-10)\) and \((-4,2)\), the midpoint is \( \left(\frac{6+(-4)}{2}, \frac{-10+2}{2}\right) = (1, -4) \).

Step 6: Calculate Slope (b)

The slope for part (b) is \( m = \frac{2 - (-10)}{-4 - 6} = \frac{12}{-10} = -\frac{6}{5} \).

Step 7: Find Negative Reciprocal Slope (b)

The perpendicular slope for part (b) is the negative reciprocal \( m = \frac{5}{6} \).

Step 8: Equation of Perpendicular Bisector (b)

Substitute the midpoint \((1, -4)\) and the slope \(\frac{5}{6}\) into \(y - y_1 = m(x - x_1)\): \[y + 4 = \frac{5}{6}(x - 1)\]. Expand to standard form: \[5x - 6y = 29\].

Step 9: Calculate Midpoint (c)

For part (c), points \((-7,-3)\) and \((5,9)\), the midpoint is \( \left(\frac{-7+5}{2}, \frac{-3+9}{2}\right) = (-1, 3) \).

Step 10: Calculate Slope (c)

The slope for part (c) is \( m = \frac{9 - (-3)}{5 - (-7)} = \frac{12}{12} = 1 \).

Step 11: Find Negative Reciprocal Slope (c)

The perpendicular slope becomes \( m = -1 \) for part (c).

Step 12: Equation of Perpendicular Bisector (c)

Substitute midpoint \((-1, 3)\) and slope \(-1\) into \(y - y_1 = m(x - x_1)\): \[y - 3 = -(x + 1)\]. Expand to standard form: \[x + y = 2\].

Step 13: Calculate Midpoint (d)

For part (d), points \((0,4)\) and \((12,-4)\), midpoint is \( \left(\frac{0+12}{2}, \frac{4+(-4)}{2}\right) = (6, 0) \).

Step 14: Calculate Slope (d)

The slope for part (d) is \( m = \frac{-4 - 4}{12 - 0} = -\frac{2}{3} \).

Step 15: Find Negative Reciprocal Slope (d)

The perpendicular slope is \( m = \frac{3}{2} \) for part (d).

Step 16: Equation of Perpendicular Bisector (d)

Substitute the midpoint \((6, 0)\) and slope \(\frac{3}{2}\) into \(y - y_1 = m(x - x_1)\): \[y - 0 = \frac{3}{2}(x - 6)\]. Expand to standard form: \[3x - 2y = 18\].

Key Concepts

Midpoint Formula

The midpoint formula helps us find the center point between two points on a plane. It's crucial in the study of geometry, especially when dealing with line segments and their bisectors.
To locate the midpoint, you simply take the average of the x-coordinates and y-coordinates of the two points. The formula is as follows:

• Given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint \( M \) is given by: \\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

For instance, the midpoint of the points \(-1, 2\) and \(3, 0\) is calculated as:

• \[ M = \left(\frac{-1+3}{2}, \frac{2+0}{2}\right) = (1, 1) \]

This point (1,1) is where the perpendicular bisector of the segment connecting these two points will pass.

Slope of a Line

Understanding the slope of a line is crucial as it shows how steep a line is. The slope is a number that describes the direction and the steepness of a line.

• The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

In identifying the slope for a line segment connecting points \((-1, 2)\) and \((3, 0)\), we find:

• Substituting the values:\[m = \frac{0 - 2}{3 - (-1)} = -\frac{1}{2}\]

This slope means that for each unit increase in \( x \), \( y \) decreases by half a unit. The slope gives us the rise over run, which is instrumental in graphing and applications in real-world situations.

Negative Reciprocal

A crucial concept in determining the equation of a perpendicular bisector is understanding the negative reciprocal. It's essential when you need to find a line perpendicular to a given one.

• Finding the negative reciprocal involves flipping the fraction of the slope and changing its sign.

For example, if a line has a slope of \(-\frac{1}{2}\), the slope of a line perpendicular to it will be:

• Negative reciprocal:\[2 = \frac{1}{-\frac{1}{2}} = 2\]

Why is this important? When two lines are perpendicular, their slopes are negative reciprocals of each other. This property is very useful in geometry and is utilized extensively in solving problems involving perpendicular lines.

Equation of a Line

The equation of a line is how we express the linear relationship between two variables in mathematical terms. It's the representation of the line on a Cartesian plane.

• The point-slope form of a line's equation, especially useful in these exercises, is given by:\[y - y_1 = m(x - x_1)\]

To determine the equation of a perpendicular bisector, after finding the midpoint \((x_1, y_1)\) and the perpendicular slope \(m\), substitute them into the point-slope form.As an illustration, with a midpoint \((1, 1)\) and perpendicular slope 2:

• Using point-slope form:\[y - 1 = 2(x - 1)\]
• Rearrange to standard form:\[2x - y = 1\]

The above equation represents the entire line on the graph. Transitioning from point-slope to standard form allows a clear comparison and understanding of different lines relative to each other, which is often needed in problem-solving.