Properties of Perpendicular Lines
Some of the common properties of perpendicular lines include:
- These lines always intersect at a 90° angle.
- Perpendicular lines intersect at a point, forming four right angles where they meet.
- The slopes of perpendicular lines are negative reciprocals of each other i.e., slope of perpendicular lines are m and -1/m.
- Product of the slopes of two perpendicular lines equals -1.
- Perpendicular lines are also referred to as orthogonal lines.
- Perpendicular lines exhibit symmetry about their point of intersection.
Equation of Perpendicular Lines
Equation of a Line with slope m and intercept c is y = mx + c. As we know, that the product of slope of perpendicular lines is -1. Thus, -1/m is the slope of line perpendicular to the given line. Using this we can easily find the equation of perpendicular lines.
Let’s consider an example for the same.
Example: Two perpendicular lines intersects at (0, 1). If equation of one line is y = 3x + 2, then find the equation of other line.
Solution:
Given: Equation of line y = 3x + 2.
Slope = 3
Thus, slope of line perpendicular to this line = -1/3
Equation of line passing through (x1, y1) with slope m is y − y1 = m(x − x1).
⇒ y − 1 = (-1/3)(x − 0)
⇒ 3(y − 1 )= -(x − 0)
⇒ 3y − 3 = -x
⇒ x + 3y − 3 = 0
⇒ x + 3y = 3
Thus, equation of the required perpendicular line is x + 3y = 3.
Parallel and Perpendicular Lines
Parallel and Perpendicular Lines are two sides of one coin. Perpendicular lines are intersecting lines, whereas parallel lines never intersect. Parallel lines in geometry are lines that never intersect and are always at the same distance from each other. On the other hand, perpendicular lines are lines that intersect each other at a right angle, forming a 90° angle. In this article, we will discuss these in detail, including examples and differences.