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Parallel and Perpendicular Lines
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Foundational Skills
State gradient of parallel line
\[ m = 4 \rightarrow m_{\parallel} = \text{?} \]
Parallel lines have equal gradients.
Perpendicular gradient (positive integer)
\[ m = 3 \rightarrow m_{\perp} = -\frac{1}{3} \]
Negative reciprocal of positive integer.
Perpendicular gradient (negative integer)
\[ m = -2 \rightarrow m_{\perp} = \frac{1}{2} \]
Negative reciprocal of negative integer.
Perpendicular gradient (fraction)
\[ m = \frac{1}{2} \rightarrow m_{\perp} = -2 \]
Negative reciprocal of a fraction.
Identifying from Equations
Verify if two equations are parallel
\[ y = 2x + 5 \text{ and } y = 2x – 3 \]
Confirm parallel by comparing gradients.
Verify if two equations are perpendicular
\[ y = 3x + 1 \text{ and } y = -\frac{1}{3}x + 2 \]
Check if gradients multiply to -1.
Identifying from Graphs
Identify parallel lines from graph
\[ \text{Are these lines parallel?} \]
Determine from a graph if lines are parallel.
Identify perpendicular lines from graph
\[ \text{Are these lines perpendicular?} \]
Determine from a graph if lines are perpendicular.
Equation of parallel line from graph
\[ \text{Parallel through } (0, 4) \]
Find equation using gradient from graph.
Equation of perpendicular line from graph
\[ \text{Perpendicular through } (0, 2) \]
Find perpendicular equation from graph.
Finding Equations
Parallel line through y-intercept
\[ \text{Parallel to } y = 2x + 3 \text{ via } (0, 5) \]
Find equation when y-intercept is given.
Parallel line through general point
\[ \text{Parallel to } y = 3x + 1 \text{ via } (2, 10) \]
Use gradient and point to find c.
Parallel line (negative gradient)
\[ \text{Parallel to } y = -2x + 4 \]
Keep the gradient negative!
Perpendicular line through y-intercept
\[ \text{Perp to } y = 4x + 2 \text{ via } (0, 3) \]
Find perpendicular when y-intercept is given.
Perpendicular line through general point
\[ \text{Perp to } y = 2x + 1 \text{ via } (4, 3) \]
Use negative reciprocal and point.
Perpendicular line (negative original)
\[ \text{Perp to } y = -3x + 2 \]
Perpendicular gradient will be positive.
Perpendicular line (fractional original)
\[ \text{Perp to } y = \frac{1}{2}x + 4 \]
Perpendicular gradient is an integer.
Applications
Equation of parallel line (word problem)
\[ \text{Line } L \text{ parallel to…} \]
Apply parallel line skills in context.
Equation of perpendicular line (word problem)
\[ \text{Line } M \text{ perpendicular to…} \]
Apply perpendicular line skills in context.
Find where perpendicular lines meet
\[ \text{Intersection point} \]
Find intersection of line and its perpendicular.
Right angle in triangle using gradients
\[ A(0,0), B(3,6), C(6,0) \]
Check for right angles using perpendicularity.
Extensions
Perpendicular bisector
\[ \text{From } A(0, 2) \text{ to } B(4, 6) \]
Find midpoint and perpendicular gradient.
Timer (Optional)
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