Determine whether the three points are collinear. (-7,4),(6,-2),(-1,1)

Verified step by step guidance

1. Recall that three points are collinear if the slope between any two pairs of points is the same. We will find the slopes between the pairs of points and compare them.
2. Calculate the slope between the first two points (-7, 4) and (6, -2) using the slope formula: \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the values: \(\frac{-2 - 4}{6 - (-7)}\).
3. Calculate the slope between the second two points (6, -2) and (-1, 1) using the same slope formula: \(\frac{1 - (-2)}{-1 - 6}\).
4. Calculate the slope between the first and third points (-7, 4) and (-1, 1) using the slope formula: \(\frac{1 - 4}{-1 - (-7)}\).
5. Compare the three slopes calculated. If all three slopes are equal, then the points are collinear; if not, they are not collinear.

Key Concepts

Collinearity of Points

Three points are collinear if they lie on the same straight line. This means the slope between any two pairs of points must be equal. Checking collinearity involves comparing slopes or using the area of the triangle formed by the points.

Slope Formula

The slope between two points (x₁, y₁) and (x₂, y₂) is calculated as (y₂ - y₁) / (x₂ - x₁). It measures the steepness of the line connecting the points. Equal slopes between pairs of points indicate they lie on the same line.

Using Area to Test Collinearity

The area of a triangle formed by three points can be found using a determinant formula. If the area is zero, the points are collinear. This method provides an alternative to slope comparison, especially when dealing with vertical lines.