How to graph a line perpendicular to another given its slope and a point

📚 Understanding Perpendicular Lines

In geometry, perpendicular lines are lines that intersect at a right angle (90 degrees). The relationship between their slopes is key to graphing them. If a line has a slope of $m$, a line perpendicular to it will have a slope of $-\frac{1}{m}$. This is known as the negative reciprocal.

📜 Historical Context

The concept of perpendicularity has been around since ancient times, with early applications in architecture and surveying. Euclid's Elements laid the foundation for understanding geometric relationships, including perpendicular lines.

📐 Key Principles

• ➕ Slope of the Given Line: Identify the slope ($m$) of the original line.
• 🔄 Negative Reciprocal: Calculate the negative reciprocal of the slope, which will be $-\frac{1}{m}$. This is the slope of the perpendicular line.
• 📍 Point-Slope Form: Use the point-slope form of a line, $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point and $m$ is the negative reciprocal slope.
• 📈 Graphing: Plot the given point and use the slope to find other points on the perpendicular line. Connect the points to draw the line.

✍️ Example Problem

Let's say we want to graph a line perpendicular to $y = 2x + 3$ that passes through the point (2, 1).

1. The slope of the given line is 2.
2. The negative reciprocal of 2 is $-\frac{1}{2}$.
3. Using the point-slope form: $y - 1 = -\frac{1}{2}(x - 2)$.
4. Simplifying, we get $y = -\frac{1}{2}x + 2$.
5. Plot the point (2, 1) and use the slope $-\frac{1}{2}$ to find other points. For example, move 2 units to the right and 1 unit down.

💡 Conclusion

Graphing a line perpendicular to another involves finding the negative reciprocal of the original line's slope and using a given point to define the new line. By understanding these steps, you can easily graph perpendicular lines.