Determine whether the three points are collinear. (0, 9),(-3, -7),(2, 19)

Verified step by step guidance

1. Recall that three points are collinear if the slope between any two pairs of points is the same.
2. Calculate the slope between the first two points (0, 9) and (-3, -7) using the formula \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\), which becomes \(\frac{-7 - 9}{-3 - 0}\).
3. Calculate the slope between the second two points (-3, -7) and (2, -19) using the same slope formula: \(\frac{-19 - (-7)}{2 - (-3)}\).
4. Compare the two slopes calculated. If they are equal, the points are collinear; if not, they are not collinear.
5. Optionally, calculate the slope between the first and third points (0, 9) and (2, -19) to confirm consistency.

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 Coordinates to Test Collinearity

To test if points are collinear, calculate the slope between the first and second points and between the second and third points. If these slopes are equal, the points are collinear. Alternatively, the area of the triangle formed by the points can be checked to be zero.