Final answer:
By calculating the distances between points using the Distance Formula, we can determine that neither set of points is collinear. The first set does not form a straight line as the distances satisfy the Pythagorean theorem, indicating a right triangle. The second set's distances do not follow the Pythagorean theorem, also showing a lack of collinearity.
Step-by-step explanation:
To determine if sets of points are collinear, we can use the Distance Formula, which is derived from the Pythagorean theorem. For points A(x1, y1) and B(x2, y2), the Distance Formula is √((x2-x1)² + (y2-y1)²). Calculating the distances between the points A to B, B to C, and A to C for each set, we can identify if these distances maintain a certain relationship to discern collinearity.
For the first set of points {A(2, 3), B(2,6), C(6,3)}, the distances AB, BC, and AC are:
- AB = √((2-2)² + (6-3)²) = 3
- BC = √((6-2)² + (3-6)²) = 5
- AC = √((6-2)² + (3-3)²) = 4
For the second set of points {A(8,3), B(5,2), C(2, 1)}, the distances are:
- AB = √((5-8)² + (2-3)²) = √9 + 1 = √10
- BC = √((2-5)² + (1-2)²) = √9 + 1 = √10
- AC = √((2-8)² + (1-3)²) = √36 + 4 = √40
Looking at these distances, for the first set of points, the lengths AB and AC relate as legs of a right triangle with BC as the hypotenuse, satisfying Pythagorean theorem (3² + 4² = 5²). Thus, points A, B, and C do not form a straight line and are not collinear. Conversely, for the second set, the distances do not satisfy the Pythagorean theorem, as (√10)² + (√10)² is not equal to (√40)². Therefore, these points are also not collinear.