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Chapter 0: Problem 91

Proof Prove that if the slopes of two nonvertical lines are negative reciprocals of each other, then the lines are perpendicular.

Short Answer

Expert verified
Assuming that the slopes of two lines are negative reciprocals of each other, it can be proved that the angles they individually make with the positive direction of x-axis sum up to 90 degrees. This leads to the conclusion that the lines are perpendicular.

Step by step solution

01

Understand the Definition of Perpendicular Lines

Two lines are said to be perpendicular if the angle between them is 90 degrees.
02

Define the Slope of a Line

The slope of a line is the ratio of the vertical change to the horizontal change between any two points on the line. It is usually represented by the letter m.
03

Understand the Concept of Negative Reciprocals

A number is said to be the negative reciprocal of another if it is the negative of the reciprocal of that number. If a and b are two slopes such that a = -1/b, then a and b are negative reciprocals of each other.
04

Combine the Definitions

Let's assume the slope of the first line is m1 and the slope of the second line is m2. If m1 and m2 are negative reciprocals of each other, then m1 = -1/m2. Using the definition of the slope being the tangent of the angle the line makes with the positive direction of the x-axis, let's assume that the angle for m1 is 胃1 and for m2 is 胃2. Then, m1 = tan(胃1) and m2 = tan(胃2). So, tan(胃1) = -1/tan(胃2). But from trigonometry, we know that tan(90 - 伪) = 1/tan(伪). So, if tan(胃1) = -1/tan(胃2), it means that 胃1 = 90 - 胃2. This shows that the lines are perpendicular to each other.

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