7.2 Parallel Lines and Perpendicular Lines
In everyday life, parallel lines and perpendicular lines appear in structures such as lane dividers in swimming pools, construction frameworks, and road intersections. In coordinate geometry, the relationship between two straight lines can be determined by analysing their gradients. The gradient of a line is directly related to the tangent of the angle it makes with the positive \(x\)-axis. By understanding how the gradients of two lines relate when they are parallel (\(m_1 = m_2\)) or perpendicular (\(m_1 m_2 = -1\)), we can determine unknown coordinates, find equations of normal or tangent lines, and solve real-world coordinate geometry problems.
Learning Objectives
- Make and verify conjectures about the relationship between the gradients of two parallel lines.
- Make and verify conjectures about the relationship between the gradients of two perpendicular lines.
- Identify whether two lines are parallel (\(m_1 = m_2\)) or perpendicular (\(m_1 m_2 = -1\)) using their equations or coordinates.
- Determine the value of an unknown constant in equations of parallel or perpendicular lines.
- Solve problems involving the equations of parallel and perpendicular lines, including perpendicular bisectors, tangents to circles, shortest distance, and intersections.
Gradient Conditions: Parallel vs Perpendicular Lines
Compare the geometric orientation, angle relationships, and gradient formulas for parallel and perpendicular lines.
Parallel Lines
Two parallel lines slope in the same direction and make equal corresponding angles with the positive \(x\)-axis: \(\theta_1 = \theta_2\).
Perpendicular Lines
Two perpendicular lines cross at a \(90^\circ\) right angle, so \(\theta_2 = 90^\circ + \theta_1\).
Short Explanation
Convert equations into gradient form, then compare the gradients using the parallel or perpendicular condition.
1. Definition and Behaviour of Gradient
The gradient \(m\) of a straight line \(L\) passing through two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is:
\[m = \dfrac{y_2 - y_1}{x_2 - x_1}\]
In \(\triangle ABC\), \(\dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{BC}{AC} = \tan \theta\). Therefore, the definition of gradient \(m\) is:
\[m = \tan \theta\]
where \(\theta\) is the angle formed between the straight line and the positive \(x\)-axis, with \(0^\circ \le \theta < 180^\circ\).
2. Sign and Value of Gradient
When \(\theta = 0^\circ\): line is horizontal, \(\tan 0^\circ = 0 \implies m = 0\).
When \(0^\circ < \theta < 90^\circ\) (acute): \(\tan \theta > 0 \implies m > 0\) (positive slope).
When \(\theta = 90^\circ\): line is vertical, \(\tan 90^\circ\) is not defined \(\implies m\) is undefined.
When \(90^\circ < \theta < 180^\circ\) (obtuse): \(\tan \theta < 0 \implies m < 0\) (negative slope).
3. Parallel Lines (\(L_1 \parallel L_2\))
If two straight lines \(L_1\) and \(L_2\) are parallel, they make equal corresponding angles with the positive \(x\)-axis:
\[\theta_1 = \theta_2 \implies \tan \theta_1 = \tan \theta_2 \implies m_1 = m_2\]
Condition: Two straight lines \(L_1\) and \(L_2\) are parallel to each other if and only if:
\[m_1 = m_2\]
4. Perpendicular Lines (\(L_1 \perp L_2\))
Let \(L_1\) and \(L_2\) be perpendicular lines intersecting at \(90^\circ\), with inclination angles \(\theta_1\) and \(\theta_2\) respectively (\(\theta_1 \ne 0^\circ\)).
Using the exterior angle of a triangle:
\[\theta_2 = 90^\circ + \theta_1\]
Taking tangent on both sides:
\[\tan \theta_2 = \tan(90^\circ + \theta_1) = -\dfrac{1}{\tan \theta_1}\]
\[\tan \theta_1 \tan \theta_2 = -1\]
Condition: Two straight lines \(L_1\) and \(L_2\) are perpendicular to each other if and only if:
\[m_1 m_2 = -1 \quad \left(\text{or } m_2 = -\dfrac{1}{m_1}\right)\]
5. Forms of Straight Line Equations Used
Gradient Form: \(y = mx + c\), where \(m\) is the gradient and \(c\) is the \(y\)-intercept.
Intercept Form: \(\dfrac{x}{a} + \dfrac{y}{b} = 1\), where \(a\) is the \(x\)-intercept, \(b\) is the \(y\)-intercept, and the gradient is:
\[m = -\dfrac{b}{a}\]
Point-Gradient Form: Equation of a line passing through \((x_1, y_1)\) with gradient \(m\):
\[y - y_1 = m(x - x_1)\]
6. Solving Geometric and Real-World Problems
Perpendicular Bisector: Passes through the midpoint of a line segment with gradient \(m_{\perp} = -\dfrac{1}{m_{\text{segment}}}\).
Tangent to a Circle: The tangent at point \(B\) is perpendicular to the radius \(AB\) at that point of contact (\(m_{\text{tangent}} \times m_{\text{radius}} = -1\)).
Shortest Distance: The shortest distance from a point to a straight line is along the perpendicular line connecting the point to the line.
Points of Intersection: Found by solving the simultaneous equations of the two intersecting lines.
| Condition | Gradient Relationship | Interpretation |
|---|---|---|
| Parallel lines | \(m_1 = m_2\) | Equal angles of inclination; lines never meet. |
| Perpendicular lines | \(m_1 m_2 = -1\) | Lines intersect at \(90^\circ\). |
| Horizontal line | \(m = 0\) | \(\theta = 0^\circ\). |
| Vertical line | Gradient is undefined | \(\theta = 90^\circ\). |
Steps to Determine Whether Two Lines Are Parallel or Perpendicular
Convert each equation into the gradient form \(y = mx + c\), or read the intercepts directly from \(\dfrac{x}{a} + \dfrac{y}{b} = 1\).
Identify the gradient of each line: \(m_1\) and \(m_2\).
If \(m_1 = m_2\), the lines are parallel.
If \(m_1 m_2 = -1\), the lines are perpendicular.
Parallel Lines Condition
\(m_1 = m_2\)Equal gradients mean equal angles of inclination.
Perpendicular Lines Condition
\(m_1 m_2 = -1\)Equivalent to \(m_2 = -\dfrac{1}{m_1}\).
Try to Answer First
Answer in your mind, then press "Check Answer".
Determine whether the lines \(2x + 3y = 9\) and \(4x + 6y = 0\) are parallel or perpendicular.
The straight line \(y = 3px - 1\) is parallel to the line \(2y = 10 - x\). Find the value of the constant \(p\).
The points \(A(1, 1)\), \(B(-1, 4)\), and \(C(5, a)\) are vertices of a triangle. If \(AB\) is perpendicular to \(BC\), find the value of \(a\).
Common Mistakes
- Forgetting to rearrange into gradient form: reading the coefficient of \(x\) as the gradient directly from \(ax + by = c\) without dividing by \(b\), mistakenly taking \(m = a\) instead of \(m = -\dfrac{a}{b}\).
- Sign error in negative reciprocal: setting \(m_2 = \dfrac{1}{m_1}\) instead of \(m_2 = -\dfrac{1}{m_1}\) for perpendicular lines.
- Incorrect intercept form gradient: given \(\dfrac{x}{a} + \dfrac{y}{b} = 1\), forgetting the negative sign and writing \(m = \dfrac{b}{a}\) instead of \(m = -\dfrac{b}{a}\).
- Confusing parallel and perpendicular conditions: using \(m_1 m_2 = -1\) for parallel lines or \(m_1 = m_2\) for perpendicular lines.
Concept Misunderstandings
Lines that intersect are always perpendicular.
Lines can intersect at any angle. They are only perpendicular if they meet specifically at \(90^\circ\), which requires \(m_1 m_2 = -1\).
The gradient is undefined for horizontal lines.
A horizontal line has \(\theta = 0^\circ\) and \(\tan 0^\circ = 0\), so its gradient is \(m = 0\). It is a vertical line (\(\theta = 90^\circ\)) whose gradient is undefined because division by zero occurs (\(\tan 90^\circ\) is undefined).
Perpendicular bisector only means perpendicular.
A perpendicular bisector has two properties: it is perpendicular to the line segment (\(m_1 m_2 = -1\)) and it passes through the midpoint of that segment.
Summary
- Gradient definition: \(m = \tan \theta\), where \(\theta\) is the angle formed with the positive \(x\)-axis (\(0^\circ \le \theta < 180^\circ\)).
- Parallel lines (\(L_1 \parallel L_2\)): Have equal angles of inclination (\(\theta_1 = \theta_2\)) and equal gradients: \(m_1 = m_2\).
- Perpendicular lines (\(L_1 \perp L_2\)): Meet at \(90^\circ\) (\(\theta_2 = 90^\circ + \theta_1\)) and the product of their gradients is \(-1\): \(m_1 m_2 = -1 \quad \Longleftrightarrow \quad m_2 = -\dfrac{1}{m_1}\).
- Line equations: Gradient form: \(y = mx + c\); Intercept form: \(\dfrac{x}{a} + \dfrac{y}{b} = 1 \implies m = -\dfrac{b}{a}\); Point-gradient form: \(y - y_1 = m(x - x_1)\).
Short Activity
Objective: Identify gradient conditions for parallel and perpendicular lines, and match geometric descriptions with their formulas.
A. Objective Quiz
1 What is the gradient of a straight line that is perpendicular to the straight line \(3x + 5y = 15\)?
2 Two lines have equations \(\dfrac{x}{3} + \dfrac{y}{9} = 1\) and \(ky = 2x - 7\). If the two lines are perpendicular to each other, what is the value of the constant \(k\)?
B. Fill in the Blanks
3 Two non-vertical straight lines \(L_1\) and \(L_2\) are parallel to each other if and only if their gradients satisfy the condition ________.
4 If a straight line forms an angle of \(\theta\) with the positive \(x\)-axis, its gradient \(m\) is defined by the trigonometric ratio \(m = \) ________.
C. Matching / Drag and Drop
Match each geometric condition or line representation with its correct mathematical gradient relationship or formula. On a phone, tap a term first, then tap the matching box.
