Final answer:
To confirm that P(2, 7) lies on the same line as Q(7, 4) and R(12, 1), we check for consistency in slopes between all pairs of points. The calculated slopes are all equal to -3/5, proving that P, Q, and R are collinear.
Step-by-step explanation:
To determine if point P(2, 7) is on the same line as points Q(7, 4) and R(12, 1), we need to check if the slope between Q and R is the same as the slope between Q and P, as well as between P and R. The slope of a line through two points (x₁, y₁) and (x₂, y₂) is calculated using the formula (y₂ - y₁) / (x₂ - x₁).
First, we calculate the slope between Q and R:
Slope of QR = (1 - 4) / (12 - 7) = -3 / 5
Then, we calculate the slope between Q and P:
Slope of QP = (7 - 4) / (2 - 7) = 3 / -5 = -3 / 5
And lastly, the slope between P and R:
Slope of PR = (1 - 7) / (12 - 2) = -6 / 10 = -3 / 5
Since the slopes of QP, QR, and PR are all the same (-3 / 5), we can conclude that points P, Q, and R lie on the same line.