Problem 5 - Algebra for College Students

Question

What is the relationship between the slopes of perpendicular lines?

Step-by-Step Solution

Answer

The slopes of perpendicular lines are negative reciprocals of each other.

Step 1 - Understand Slope

Slope is a measure of the steepness of a line. It is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line, often represented as \(m = \frac{\Delta y}{\Delta x}\).

Step 2 - Define Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (90 degrees).

Step 3 - Relationship of Slopes

The slopes of two perpendicular lines are negative reciprocals of each other. If one line has a slope of \(m_1\), then the slope of the perpendicular line \(m_2\) can be written as \(m_2 = -\frac{1}{m_1}\).

Step 4 - Verify with Example

For example, if a line has a slope of \(2\), a line perpendicular to it will have a slope of \(-\frac{1}{2}\).

Step 5 - Conclude the Relationship

Thus, if the product of the slopes of two lines is \(-1\), the lines are perpendicular, i.e., \(m_1 \cdot m_2 = -1\).

Key Concepts

Slope Calculation

To master the concept of slope, let's first understand its calculation. Slope is essentially the measure of a line's steepness or inclination. It's determined by how much the y-coordinate (vertical change) varies relative to the x-coordinate (horizontal change) between two points on a line.

The formula to calculate the slope, typically denoted as 'm', is given as: \[ m = \frac{\text{change in y}}{\text{change in x}} = \frac{\text{rise}}{\text{run}} \ \text{or} \ m = \frac{y_2 - y_1}{x_2 - x_1} \ \text{where} \ (x_1, y_1) \text{ and } (x_2, y_2) \text{are two points on the line} \] This formula helps us understand how steep or flat a line is.

For example, if you have two points on a line, say (1, 2) and (3, 6), you plug these into the formula: \[ m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 \ \text{Thus, the slope of the line is 2.} \]

Perpendicular Lines

Perpendicular lines are special because they intersect at a right angle (90 degrees). When two lines are perpendicular, their slopes have a specific relationship. Contrary to parallel lines that never meet and have the same slope, perpendicular lines do meet and have a very unique slope relationship.

Imagine two lines meeting at a right angle. If one has a steep incline, the other must have a slope that counteracts this so they meet at 90 degrees. This brings us to the next concept: negative reciprocals of slopes.

For two lines to be perpendicular, one line's slope is the negative reciprocal of the other's. This ensures that their intersection results in a right angle.

Negative Reciprocals

Understanding negative reciprocals is key when studying perpendicular lines. The concept of reciprocals in mathematics means flipping a fraction. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).

When two lines are perpendicular, not only do we use reciprocals, but one slope is the negative reciprocal of the other. This means if one line has a slope of \( m_1 = \frac{a}{b} \), the slope of the perpendicular line \( m_2 = -\frac{b}{a} \).

For example, if the slope of one line is 2, the slope of the perpendicular line must be -1/2. If you multiply these slopes, their product is always -1: \[ m_1 × m_2 = 2 × -\frac{1}{2} = -1 \] This principle helps confirm that the lines are indeed perpendicular.

Using this understanding, you can work on multiple problems involving perpendicular lines with ease!