Parallel and perpendicular lines

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 \(-1\).