If three distinct points A, B, and C in a plane are such that the slopes of nonvertical line segments AB, AC, and BC are equal, then A, B, and C are collinear. Otherwise, they are not. Use this fact to determine whether the three points given are collinear. (-1, -3), (-5, 12), (1, -11)

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Recall that three points A, B, and C are collinear if the slopes of the line segments AB, AC, and BC are all equal. This means the slope between any two pairs of points must be the same.

Label the points as A(-1, -3), B(-5, 12), and C(1, -11). We will calculate the slopes of AB, AC, and BC using the slope formula: \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\).

Calculate the slope of AB: \(m_{AB} = \frac{12 - (-3)}{-5 - (-1)} = \frac{12 + 3}{-5 + 1} = \frac{15}{-4}\).

Calculate the slope of AC: \(m_{AC} = \frac{-11 - (-3)}{1 - (-1)} = \frac{-11 + 3}{1 + 1} = \frac{-8}{2}\).

Calculate the slope of BC: \(m_{BC} = \frac{-11 - 12}{1 - (-5)} = \frac{-23}{6}\). Compare \(m_{AB}\), \(m_{AC}\), and \(m_{BC}\). If all three are equal, the points are collinear; otherwise, they are not.

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주요 개념

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Slope of a Line Segment

The slope measures the steepness of a line segment connecting two points and is calculated as the change in y-coordinates divided by the change in x-coordinates. It indicates how much y changes for a unit change in x. For points (x1, y1) and (x2, y2), slope = (y2 - y1) / (x2 - x1), provided the line is not vertical.

Collinearity of Points

Three points are collinear if they lie on the same straight line. This occurs when the slopes of the line segments connecting each pair of points are equal. If the slopes between AB, AC, and BC are the same, the points A, B, and C are collinear.

Using Slopes to Test Collinearity

To determine if three points are collinear, calculate the slopes of the segments AB, AC, and BC. If all slopes are equal (and none are undefined due to vertical lines), the points lie on the same line. Otherwise, they are not collinear.