Parallel & Perpendicular Lines Calculator
Algebra, Lines
What are parallel and perpendicular lines
Definition: Parallel lines have the same slope. Perpendicular lines meet at a right angle and their slopes multiply to -1 when both slopes are defined.
Formula: $$m_{\parallel}=m,\quad m_{\perp}=-\frac{1}{m}$$
Intro: This perpendicular line calculator finds the equation of the line parallel or perpendicular to your given line through your point. It also works as a perpendicular gradient calculator.
Accepted line forms
- Slope–intercept: $y=mx+b$
- Standard form: $Ax+By=C$ ( $m=-A/B$ )
- Vertical line: $x=c$ ( $perpendicular\;is\;y=\text{constant}$ )
How this calculator works
- Enter your given line in a common form like
y = (2/3)x - 1or2x + 3y = 6. - Enter the point
(x0, y0). - Choose Parallel or Perpendicular.
- MathGPT extracts the slope, compute the target slope, then build the line through your point using point slope form.
Worked example
- Line $y=\tfrac{2}{3}x - 1$; find the perpendicular through $(3,4)$.
- Identify slope of the given line in $y=mx+b$ form: $$m=\tfrac{2}{3}.$$
- Perpendicular slope is the negative reciprocal: $$m_{\perp}=-\frac{1}{m}=-\frac{1}{\tfrac{2}{3}}=-\tfrac{3}{2}.$$
- Use point–slope form with point $(x_0,y_0)=(3,4)$: $$y-y_0=m_{\perp}(x-x_0)\;\Rightarrow\; y-4=-\tfrac{3}{2}(x-3).$$
- Distribute and solve for $y$: $$y-4=-\tfrac{3}{2}x+\tfrac{9}{2} \;\Rightarrow\; y=-\tfrac{3}{2}x+\tfrac{9}{2}+4.$$
- Combine constants (write $4=\tfrac{8}{2}$): $$y=-\tfrac{3}{2}x+\tfrac{9}{2}+\tfrac{8}{2}=-\tfrac{3}{2}x+\tfrac{17}{2}.$$
- Final perpendicular line through $(3,4)$: $$\boxed{y=-\tfrac{3}{2}x+\tfrac{17}{2}}.$$
- Quick check: Slope product $m\cdot m_{\perp}=\tfrac{2}{3}\cdot(-\tfrac{3}{2})=-1$ ✔️; point check: plug $x=3$ gives $y=-\tfrac{9}{2}+\tfrac{17}{2}=\tfrac{8}{2}=4$ ✔️.
- Line $y=-\tfrac{5}{4}x+7$; find the parallel line through $(−2,1)$.
- Given slope: $$m=-\tfrac{5}{4}.$$
- Parallel lines have the **same** slope: $$m_{\parallel}=m=-\tfrac{5}{4}.$$
- Point–slope with $(x_0,y_0)=(-2,1)$: $$y-1=-\tfrac{5}{4}(x-(-2))=-\tfrac{5}{4}(x+2).$$
- Expand and solve for $y$: $$y-1=-\tfrac{5}{4}x-\tfrac{5}{2}\;\Rightarrow\; y=-\tfrac{5}{4}x-\tfrac{5}{2}+1.$$
- Combine constants (write $1=\tfrac{2}{2}$): $$y=-\tfrac{5}{4}x-\tfrac{5}{2}+\tfrac{2}{2}=-\tfrac{5}{4}x-\tfrac{3}{2}.$$
- Final parallel line through $(-2,1)$: $$\boxed{y=-\tfrac{5}{4}x-\tfrac{3}{2}}.$$
- Check the point: $x=-2 \Rightarrow y=-\tfrac{5}{4}(-2)-\tfrac{3}{2}=\tfrac{10}{4}-\tfrac{3}{2}=\tfrac{5}{2}-\tfrac{3}{2}=1$ ✔️.
- Line $x=4$; find a perpendicular through $(3,4)$.
- A vertical line $x=4$ has undefined slope.
- Perpendicular to a vertical line is **horizontal**: slope $0$, equation $y=\text{constant}$.
- Horizontal line through $(3,4)$ is $$\boxed{y=4}.$$
- Parallel to $x=4$ through $(3,4)$ would be another vertical line: $$\boxed{x=3}.$$
- Line $3x + 4y = 12$; find the perpendicular through $(2,-1)$.
- Convert standard form $Ax+By=C$ to slope-intercept by solving for $y$: $$3x+4y=12\;\Rightarrow\; 4y=-3x+12\;\Rightarrow\; y=-\tfrac{3}{4}x+3.$$
- Identify the slope: $$m=-\tfrac{3}{4}.$$
- Perpendicular slope is the negative reciprocal: $$m_{\perp}=-\frac{1}{m}=-\frac{1}{-\tfrac{3}{4}}=\tfrac{4}{3}.$$
- Use point–slope form with $(x_0,y_0)=(2,-1)$: $$y-(-1)=\tfrac{4}{3}(x-2)\;\Rightarrow\; y+1=\tfrac{4}{3}(x-2).$$
- Distribute and solve for $y$: $$y+1=\tfrac{4}{3}x-\tfrac{8}{3}\;\Rightarrow\; y=\tfrac{4}{3}x-\tfrac{8}{3}-1.$$
- Combine constants (write $1=\tfrac{3}{3}$): $$y=\tfrac{4}{3}x-\tfrac{8}{3}-\tfrac{3}{3}=\tfrac{4}{3}x-\tfrac{11}{3}.$$
- Final perpendicular line through $(2,-1)$: $$\boxed{y=\tfrac{4}{3}x-\tfrac{11}{3}}.$$
- Verify: Slope product $(-\tfrac{3}{4})\cdot(\tfrac{4}{3})=-1$ ✔️; point check: $x=2$ gives $y=\tfrac{8}{3}-\tfrac{11}{3}=-\tfrac{3}{3}=-1$ ✔️.
- Line $3x+4y=12$; find parallel through $(0,0)$ and distance between them.
- Convert to slope-intercept: $$y=-\tfrac{3}{4}x+3$$ so $$m=-\tfrac{3}{4}.$$
- Parallel has same slope through $(0,0)$: $$y-0=-\tfrac{3}{4}(x-0)\;\Rightarrow\; y=-\tfrac{3}{4}x.$$
- Convert to standard form: $$4y=-3x\;\Rightarrow\; 3x+4y=0.$$ $$\boxed{3x+4y=0}.$$
- **Distance formula** between parallel lines $Ax+By=C_1$ and $Ax+By=C_2$: $$d=\frac{|C_2-C_1|}{\sqrt{A^2+B^2}}.$$
- Apply with $3x+4y=12$ and $3x+4y=0$: $$d=\frac{|12-0|}{\sqrt{3^2+4^2}}=\frac{12}{\sqrt{9+16}}=\frac{12}{5}.$$
- Distance between the parallel lines: $$\boxed{d=\tfrac{12}{5}=2.4\text{ units}}.$$
- Common input errors and how to fix them.
- **Error**: Line entered as $xy=5$ or $y^2=x$. **Fix**: These are not linear equations. Use $y=mx+b$, $Ax+By=C$, or $x=c$ form only.
- **Error**: Point entered as $3,4$ or $x=3,y=4$. **Fix**: Use parentheses and comma: $(3,4)$.
- **Error**: Incomplete point like $(3,)$ or $(,4)$. **Fix**: Provide both coordinates: $(3,4)$.
- **Error**: Division by zero when finding perpendicular to $y=0$. **Result**: Calculator returns vertical line $x=x_0$.
- **Error**: Mixing decimals and fractions like $y=0.5x+\tfrac{1}{2}$. **Tip**: Convert all to one format: $y=\tfrac{1}{2}x+\tfrac{1}{2}$ or $y=0.5x+0.5$.
FAQs
What if the given line is vertical?
Parallel is another vertical line $x=c$; perpendicular is horizontal $y=c$.
Do I need slope–intercept form first?
If the line is not already in $y=mx+b$, rearrange to isolate $y$ and read $m$. For vertical lines ($x=c$), treat separately as above.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
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More ways MathGPT can help
- Ask questions and get step by step explanations you can copy into your notes.
- Practice mode to generate similar parallel and perpendicular line problems.
- Create flashcards from the rules and examples. ( Flashcard )
- Visualize parallel and perpendicular lines with interactive graphing. ( Graph visualizer )
- Try Kids Mode for beginner friendly explanations, or Advanced Mode for full algebra steps. ( Example )
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- Watch video walk through examples on YouTube. ( YouTube channel )