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 $-1$.