Final answer:
The correct statement regarding the points A(1, 2), B(4, 5), and C(6, 7) is that they are collinear as they share the same slope of 1. AB and BC are neither perpendicular nor congruent.
Step-by-step explanation:
The question asks for the correct statement regarding the points A(1, 2), B(4, 5), and C(6, 7).
Let's determine the slope of AB and BC to see if they are congruent (equal in length) and/or perpendicular. The slope of a line is given by the change in y divided by the change in x (rise over run).
For AB: Slope m = (y2 - y1) / (x2 - x1) = (5 - 2) / (4 - 1) = 3 / 3 = 1.
For BC: Slope m = (y2 - y1) / (x2 - x1) = (7 - 5) / (6 - 4) = 2 / 2 = 1.
The slopes of AB and BC are equal, indicating they are parallel, not perpendicular. Now let's find the lengths of AB and BC to see if they are equal (congruent).
Length of AB = √[(x2 - x1)² + (y2 - y1)²] = √[(4 - 1)² + (5 - 2)²] = √[9 + 9] = √18.
Length of BC = √[(x2 - x1)² + (y2 - y1)²] = √[(6 - 4)² + (7 - 5)²] = √[4 + 4] = √8.
The lengths of AB and BC are not equal, so they are not congruent. Therefore, the correct statement is true B: Points A, B, and C are collinear because they lie on the same line with the same slope of 1.