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Chapter 1: Problem 69

Determine whether the points lie on a straight line. $$ A(-2,1), B(1,7), \text { and } C(4,13) $$

Short Answer

Expert verified
Yes, the points A\((-2,1)\), B\((1,7)\), and C\((4,13)\) lie on a straight line, as the slopes between A and B, and B and C are both equal to 2.

Step by step solution

01

Find the slope between points A and B

To find the slope between points A and B, use the formula: \( m_{AB} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\) Use the coordinates of A and B: \( A(-2,1) \) and \( B(1,7) \) \( m_{AB} = \frac{7 - 1}{1 - (-2)} = \frac{6}{3} = 2 \) The slope between points A and B is 2.
02

Find the slope between points B and C

To find the slope between points B and C, use the formula: \( m_{BC} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \) Use the coordinates of B and C: \( B(1,7) \) and \( C(4,13) \) \( m_{BC} = \frac{13 - 7}{4 - 1} = \frac{6}{3} = 2 \) The slope between points B and C is 2.
03

Compare the slopes

Now we will compare the slopes to determine if the points lie on a straight line. If \(m_{AB} = m_{BC}\), then A, B, and C lie on a straight line. In our case, \(m_{AB} = 2\) and \(m_{BC} = 2\) Since \(m_{AB} = m_{BC}\) , the points A, B, and C lie on a straight line.

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