Understanding Perpendicular Lines | The Relationship Between Slopes and Right Angles

In a coordinate plane, two nonvertical lines are perpendicular lines if and only if the product of their slopes is -1.

In a coordinate plane, lines can be described by their slopes

In a coordinate plane, lines can be described by their slopes. The slope of a line measures the steepness of the line and is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

Two lines are said to be perpendicular if they intersect at a right angle, forming a 90-degree angle. In other words, if one line is horizontal (slope = 0), the other line is vertical (slope is undefined), or vice versa.

Now, let’s consider two nonvertical lines, Line A and Line B. If these two lines are perpendicular, their slopes must have a specific relationship. The product of their slopes should be -1.

To understand why the product of slopes is -1 for perpendicular lines, let’s take a closer look at their slopes. Since the slope of a line is a ratio, we can represent it as a fraction m = rise/run.

Let the slope of Line A be represented as m1, and the slope of Line B as m2. If the two lines are perpendicular, we know that Line A might look something like this: m1 = -1/m2. In other words, the ratio of the rise to run for Line A is the negative reciprocal of the ratio of the rise to run for Line B.

Let’s demonstrate this with an example:
Suppose Line A has a slope of 2/3. Then, the slope of Line B, if perpendicular, should be -3/2. We can confirm this by calculating their product: (2/3) * (-3/2) = -1.

By using this concept, we can determine whether two nonvertical lines are perpendicular by finding their slopes and checking if their product is -1. If the product is not -1, then the lines are not perpendicular.

It’s important to note that this property holds true only for nonvertical lines. Vertical lines have an undefined slope, so the product of their slopes cannot be calculated.

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