Finding the Slope of a Perpendicular Line
The slope of a line measures its steepness and is usually denoted by m in the equation y = mx + c. To find the slope of a line perpendicular to a given line, first express the original line in slope-intercept form (y = mx + c). The slope of the perpendicular line is the negative reciprocal of the original slope. This means if the original slope is m, the perpendicular slope is -1/m.
For example, consider the line given by 2y = 3x + 1. To find its slope, divide both sides by 2 to get y = (3/2)x + 1/2. The slope m is 3/2. Therefore, the slope of any line perpendicular to this is -1 divided by (3/2), which equals -2/3.
- Rewrite the original line in y = mx + c form.
- Identify the slope m.
- Calculate the negative reciprocal: perpendicular slope = -1/m.
Calculating the Midpoint of a Segment
When finding the equation of a perpendicular bisector, the line must pass through the midpoint of the segment it bisects. The midpoint is the point exactly halfway between two given points (x₁, y₁) and (x₂, y₂).
The midpoint coordinates are calculated using the midpoint formula:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). This point will be used as a reference point for the perpendicular bisector.
- Add the x-coordinates and divide by 2.
- Add the y-coordinates and divide by 2.
- Use the resulting point as the midpoint.
Determining the Slope of the Segment and Its Perpendicular
Next, find the slope of the segment connecting the two points. The slope m is given by the difference in y-coordinates divided by the difference in x-coordinates:
m = (y₂ - y₁) / (x₂ - x₁). Once this slope is found, the slope of the perpendicular bisector is the negative reciprocal of this value.
For example, for points (-3, 8) and (9, -2): the slope of the segment is (-2 - 8) / (9 - (-3)) = (-10) / 12 = -5/6. The perpendicular slope is then 6/5.
- Calculate segment slope using (y₂ - y₁) / (x₂ - x₁).
- Find perpendicular slope as negative reciprocal: -1/m.
Writing the Equation of the Perpendicular Bisector
With the perpendicular slope and midpoint known, use the point-slope form of a line to write the equation:
y - y₁ = m(x - x₁), where (x₁, y₁) is the midpoint and m is the perpendicular slope.
Continuing the example with midpoint (3, 3) and slope 6/5: y - 3 = (6/5)(x - 3). This can be rearranged into slope-intercept form if desired.
- Use point-slope form: y - y₁ = m(x - x₁).
- Substitute midpoint coordinates and perpendicular slope.
- Simplify to desired form (slope-intercept or standard).
Summary and Practical Tips
Always start by rewriting any given line into y = mx + c form to identify the slope easily.
Remember the negative reciprocal relationship between slopes of perpendicular lines: if m is the original slope, perpendicular slope = -1/m.
The perpendicular bisector passes through the midpoint of the segment it bisects and has slope equal to the negative reciprocal of the segment's slope, ensuring it is perpendicular and bisects the segment exactly at its midpoint.
- Check calculations carefully, especially signs when finding slopes.
- Use exact fractions rather than decimals for accuracy.
- Label points and slopes clearly in working to avoid confusion.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Find slope of original line | Rewrite line in y = mx + c form and identify m | 2y = 3x + 1 → y = (3/2)x + 1/2, slope m = 3/2 | Essential first step to find perpendicular slope |
| Calculate perpendicular slope | Take negative reciprocal of original slope | If m = 3/2, perpendicular slope = -2/3 | Key to determining perpendicular line equation |
| Find midpoint of segment | Use midpoint formula ((x₁+x₂)/2, (y₁+y₂)/2) | Points (-3,8) and (9,-2) → midpoint (3,3) | Needed for perpendicular bisector equation |
| Calculate segment slope | Use (y₂ - y₁)/(x₂ - x₁) | Slope = (-2 - 8)/(9 - (-3)) = -10/12 = -5/6 | Used to find perpendicular slope for bisector |
| Write equation of perpendicular bisector | Use point-slope form y - y₁ = m(x - x₁) with midpoint and perpendicular slope | y - 3 = (6/5)(x - 3) | Final step to express perpendicular bisector equation |
Active recall
Quick check
1. What is the slope of a line perpendicular to y = 4x + 7?
The slope of the given line is 4. The perpendicular slope is the negative reciprocal: -1/4.
2. Find the midpoint of the segment connecting (2, 5) and (8, 1).
Midpoint = ((2 + 8)/2, (5 + 1)/2) = (10/2, 6/2) = (5, 3).
3. Calculate the slope of the segment between points (0, 0) and (4, 8).
Slope = (8 - 0) / (4 - 0) = 8/4 = 2.
4. Write the equation of the perpendicular bisector of the segment joining (1, 3) and (5, 7).
Midpoint = ((1+5)/2, (3+7)/2) = (3, 5). Slope of segment = (7-3)/(5-1) = 4/4 = 1. Perpendicular slope = -1. Equation: y - 5 = -1(x - 3), or y = -x + 8.