91Ó°ÊÓ

Perpendicular lines have a unique relationship in geometry. When two lines in a plane are perpendicular, they intersect at a 90-degree angle, forming right angles at the intersection point. This concept is crucial, especially when dealing with coordinate geometry.
In the context of linear equations, if one line's equation is known and you need to find a line perpendicular to it, you have to understand how their slopes relate to each other. That brings us to considering the slopes of these lines.

The slope of a line is a measure of its steepness. It's represented by the letter 'm' in the equation of a line given in slope-intercept form, which is written as:

• \( y = mx + b \)

The slope describes how much \( y \) increases or decreases as \( x \) increases by one unit. A positive slope indicates an upward trend, while a negative slope shows a downward one.
To find the slope from a given equation, it helps to express the equation in the standard slope-intercept form. For example, for the equation \( x + 3y = 8 \), by isolating \( y \), we find the slope to be \( -\frac{1}{3} \). Understanding the slope is essential, especially when determining relationships between lines, such as whether they are parallel or perpendicular.

The concept of the negative reciprocal is central when identifying perpendicular lines. If two lines are perpendicular, the slope of one line is the negative reciprocal of the other.
What this means is that if one line has a slope of \( m_1 \), the slope \( m_2 \) of a perpendicular line will follow the formula:

• \( m_2 = -\frac{1}{m_1} \)

When considering our example with the slope \(-\frac{1}{3}\), the slope of any line perpendicular to it would be \( 3 \). Simply flip the original slope to become \( \frac{3}{1} \) and change its sign to derive the perpendicular slope.

The point-slope form is a powerful tool for writing the equation of a line when you know its slope and a point through which it passes. The point-slope form is structured as:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) denotes a specific point on the line, while \( m \) is the slope.
This form is particularly useful because it allows you to plug in the known values and derive a line's equation quickly. For a line passing through the point \((-2, 2)\) with a slope of \( 3 \), the equation is:

• \( y - 2 = 3(x + 2) \)

This equation can then be simplified to reach the familiar slope-intercept form.