Perpendicular Slope Calculator
Finds the line perpendicular to (or parallel with) a given equation that passes through a specified point, returning slope and equation.
Inputs
Equation Preview
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Helping Notes
y = mx + b, ax + by + c = 0, x = k (vertical), y = k (horizontal).Results
Given line slope:
Result slope (requested):
Equation of resulting line:
What is Perpendicular Slope Calculator
A Perpendicular Slope Calculator finds the slope of a line that is perpendicular to a given line and can produce the equation of the perpendicular line through a specified point. The core geometric identity used is that perpendicular slopes are negative reciprocals: if two lines have slopes \(m_1\) and \(m_2\) and are perpendicular, then \[ m_1 \cdot m_2 = -1, \] so the perpendicular slope is computed as \[ m_{\perp} = -\frac{1}{m}. \] The tool accepts a line (in common forms) or a slope plus a point, then applies algebraic manipulation to yield the required slope and the point-slope or slope-intercept form of the perpendicular line.
About the Perpendicular Slope Calculator
This calculator supports common input styles: a line in slope-intercept form \(y = mx + b\), a standard linear equation \(ax + by + c = 0\), or simply the numeric slope \(m\). When provided a point \((x_1,y_1)\) through which the perpendicular line must pass, the calculator computes \(m_{\perp}\) and uses the point-slope equation \[ y - y_1 = m_{\perp}(x - x_1) \] to show the perpendicular line. For convenience the calculator can convert the result into slope-intercept form \(y = m'x + b'\) by solving for the intercept \(b' = y_1 - m_{\perp} x_1\). It also handles edge cases: horizontal lines (slope \(0\)) produce vertical perpendiculars (undefined slope, equation \(x = \text{constant}\)), and vertical lines produce horizontal perpendiculars (slope \(0\)).
How to Use this Perpendicular Slope Calculator
1. Enter the given line (for example \(y = 2x + 3\) or \(x - 3y + 6 = 0\)) or directly enter the slope value \(m\).
2. Enter the point coordinates \((x_1,y_1)\) through which the perpendicular line must pass.
3. Click Calculate. The tool computes the perpendicular slope using \(m_{\perp} = -1/m\) and displays the point-slope form
\[
y - y_1 = m_{\perp}(x - x_1),
\]
and then simplifies to slope-intercept form \(y = m_{\perp} x + b\) by evaluating \(b = y_1 - m_{\perp} x_1\).
4. Use Reset to clear inputs, or example buttons to load sample problems instantly. On small screens the results are scrolled into view automatically for convenience.
Examples
Example 1: Given line \(y = 2x + 3\) and point \((4,1)\). The original slope \(m = 2\). The perpendicular slope is \(m_{\perp} = -\tfrac{1}{2}\). Point-slope: \(y - 1 = -\tfrac{1}{2}(x - 4)\). Simplified: \(y = -\tfrac{1}{2}x + 3\).
Example 2: Given vertical line \(x = 5\) and point \((5, -2)\). Vertical lines have undefined slope, so the perpendicular line is horizontal with slope \(0\): \(y = -2\).
Example 3: Given line \(3x + 4y - 12 = 0\). Rearranged: \(y = -\tfrac{3}{4}x + 3\). Original slope \(m = -\tfrac{3}{4}\). Perpendicular slope \(m_{\perp} = \tfrac{4}{3}\). Through point \((1,1)\): \(y - 1 = \tfrac{4}{3}(x - 1)\).
Frequently Asked Questions
What is the perpendicular slope of a line with slope \(m\)?
The perpendicular slope is the negative reciprocal: \(m_{\perp} = -\tfrac{1}{m}\) (provided \(m \neq 0\)).
How do I handle vertical or horizontal lines?
Horizontal lines have slope \(0\) and perpendicular lines are vertical (equation \(x = c\)). Vertical lines have undefined slope and perpendicular lines are horizontal with slope \(0\).
Can I enter a line in the form \(ax+by+c=0\)?
Yes. Rearranging to slope-intercept form yields \(y = -\tfrac{a}{b}x - \tfrac{c}{b}\), giving slope \(m = -\tfrac{a}{b}\), then compute the negative reciprocal.
What if the input slope is zero?
If the input slope is zero the perpendicular line is vertical and cannot be expressed as \(y = mx + b\); use the vertical form \(x = x_1\).
How is the perpendicular line through a point found?
After computing \(m_{\perp}\), use the point-slope formula \(y - y_1 = m_{\perp}(x - x_1)\) then simplify to the desired form.
Why is the product of perpendicular slopes \(-1\)?
In Euclidean plane geometry, perpendicular lines have slopes that multiply to \(-1\) because their direction vectors have a dot product of zero, producing the negative-reciprocal relationship.
Can I input decimals or fractions for slopes?
Yes. Decimal or fractional slopes work the same; the calculator will compute the negative reciprocal and simplify where possible.
Does the calculator return exact fractions or decimals?
Depending on implementation it may return exact rational values (fractions) or decimal approximations; both are valid representations of the perpendicular slope.
How do I find the intersection of the original and perpendicular lines?
Set the two line equations equal and solve for \(x\), then compute \(y\). If the original is vertical, handle it as a special case using \(x=\) constant.
What if my point is not on the original line?
The perpendicular line through a point does not need to intersect the original line at that point; it simply has the perpendicular slope and passes through the given point.
Can the calculator handle symbolic coefficients?
Some implementations accept symbolic coefficients (like parameters) and return symbolic perpendicular slopes; typical online calculators expect numeric inputs.
Is there a formula to convert point-slope to slope-intercept form?
Yes. From \(y-y_1=m(x-x_1)\), expand and solve for \(y\): \(y = m x + (y_1 - m x_1)\), where the intercept is \(b = y_1 - m x_1\).