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Chapter 1: Problem 16

Determine whether the lines through the pairs of points are perpendicular. $$ A(2,0), B(1,-2) \text { and } C(4,2), D(-8,4) $$

Short Answer

Expert verified
The slopes of the lines passing through A and B, and C and D are \(m_{AB} = 2\) and \(m_{CD} = -\frac{1}{6}\), respectively. Since the product of their slopes is -\(\frac{1}{3}\), which is not equal to -1, the lines are not perpendicular.

Step by step solution

01

Find the slope of the line passing through A and B

To find the slope of the line passing through A(2,0) and B(1,-2), we'll use the formula for the slope: \( m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \) Here, \(x_1 = 2, y_1 = 0, x_2 = 1, y_2 = -2\) Substitute the values into the formula: \( m_{AB} = \frac{-2 - 0}{1 - 2} = \frac{-2}{-1} = 2 \)
02

Find the slope of the line passing through C and D

To find the slope of the line passing through C(4,2) and D(-8,4), we'll use the formula for the slope: \( m_{CD} = \frac{y_2 - y_1}{x_2 - x_1} \) Here, \(x_1 = 4, y_1 = 2, x_2 = -8, y_2 = 4\) Substitute the values into the formula: \( m_{CD} = \frac{4 - 2}{-8 - 4} = \frac{2}{-12} = -\frac{1}{6} \)
03

Check if the product of the slopes is equal to -1

To check if the lines are perpendicular, we'll find the product of their slopes and check if it's equal to -1: \( m_{AB} \cdot m_{CD} = 2 \cdot -\frac{1}{6} = -\frac{2}{6} = -\frac{1}{3} \) Since the product of the slopes is not equal to -1, the lines through the pairs of points A and B, and C and D are not perpendicular.

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