Parallel and perpendicular lines

Topic summary

In geometry, parallel and perpendicular lines are fundamental concepts that describe the relationship between two lines. Parallel lines never intersect, while perpendicular lines intersect at a right angle. Understanding these relationships is essential for working with lines and gradients in coordinate geometry.

1. Parallel Lines:

Two lines are parallel if they have the same slope and never intersect. This means that the gradient (slope) of the two lines is identical. The general equation of a line is given by:

\[ y = mx + c \] where \( m \) is the slope (gradient) and \( c \) is the y-intercept. For two lines to be parallel, they must have the same value of \( m \).

Example:

Consider the equations of two lines:

• Line 1: \( y = 2x + 3 \)
• Line 2: \( y = 2x – 4 \)

Both lines have a slope of \( m = 2 \), so they are parallel. They will never intersect, as their gradients are identical.

2. Perpendicular Lines:

Two lines are perpendicular if their gradients are negative reciprocals of each other. This means that the product of their gradients is equal to \(-1\). If the slope of one line is \( m_1 \) and the slope of the other line is \( m_2 \), the condition for perpendicularity is:

\[ m_1 \times m_2 = -1 \]

Example:

Consider the equations of two lines:

• Line 1: \( y = 3x + 2 \)
• Line 2: \( y = -\frac{1}{3}x – 1 \)

The slope of Line 1 is \( m_1 = 3 \), and the slope of Line 2 is \( m_2 = -\frac{1}{3} \). Their product is:

\[ 3 \times -\frac{1}{3} = -1 \]

Since the product of the gradients is \(-1\), the two lines are perpendicular and intersect at a right angle.

3. Summary:

• Two lines are parallel if their gradients are equal.
• Two lines are perpendicular if the product of their gradients is \(-1\).