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Chapter 1: Problem 15

Determine whether the lines through the pairs of points are perpendicular. $$ A(-2,5), B(4,2) \text { and } C(-1,-2), D(3,6) $$

Short Answer

Expert verified
Since the slopes of lines AB and CD are \(m_{AB} = -\frac{1}{2}\) and \(m_{CD} = 2\), and their product is \((- \frac{1}{2}) (2) = -1\), we can conclude that the lines through pairs of points A(-2,5), B(4,2) and C(-1,-2), D(3,6) are perpendicular.

Step by step solution

01

Find the slope of line AB

To find the slope of line AB, use the slope formula with points A and B: \(m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 5}{4 - (-2)} = \frac{-3}{6}\)
02

Simplify the slope of line AB

Simplify the slope of line AB by dividing the numerator and the denominator by the greatest common divisor (3 in this case): \(m_{AB} = \frac{-3}{6} = -\frac{1}{2}\)
03

Find the slope of line CD

To find the slope of line CD, use the slope formula with points C and D: \( m_{CD} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - (-2)}{3 - (-1)} = \frac{8}{4} \)
04

Simplify the slope of line CD

Simplify the slope of line CD by dividing the numerator and the denominator by the greatest common divisor (4 in this case): \(m_{CD} = \frac{8}{4} = 2\)
05

Check if the product of slopes is -1

Since the slopes of lines AB and CD are -1/2 and 2 respectively, we will check if their product is equal to -1: \((- \frac{1}{2}) (2) = -1\)
06

Determine whether lines are perpendicular

As we found that the product of the slopes is -1, we can conclude that the lines through pairs of points A(-2,5), B(4,2) and C(-1,-2), D(3,6) are perpendicular.

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