Determine whether the three points are collinear. (0,-7),(-3,5),(2,-15)

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Recall that three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are collinear if the slope between any two pairs of points is the same.

Calculate the slope between the first two points $(0, -7)$ and $(-3, 5)$ using the formula: $slope = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-7)}{-3 - 0} = \frac{5 + 7}{-3} = \frac{12}{-3}$

Calculate the slope between the second two points $(-3, 5)$ and $(2, -15)$ using the same formula: $slope = \frac{-15 - 5}{2 - (-3)} = \frac{-15 - 5}{2 + 3} = \frac{-20}{5}$

Compare the two slopes calculated. If they are equal, the points are collinear; if not, they are not collinear.

Optionally, calculate the slope between the first and third points $(0, -7)$ and $(2, -15)$ to confirm consistency: $slope = \frac{-15 - (-7)}{2 - 0} = \frac{-15 + 7}{2} = \frac{-8}{2}$

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 the Area of a Triangle to Test Collinearity

If three points form a triangle with zero area, they are collinear. The area can be found using the determinant formula: 0.5 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|. A result of zero confirms collinearity.

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