Question

Determine whether the given points are collinear. Points are collinear if they can be labeled P, Q, and R so that (d(P,Q) + d(Q,R) = d(P,R)).

(9, -9), (16, 1), (1, 5)

Choose the correct answer below.

A. Yes because (d(P,Q) + d(Q,R) = d(P,R)), where (P = (1,5)), (Q = (16,1)), and (R=(9,-9)).
B. Yes because (d(P,Q) + d(Q,R) = d(P,R)), where (P = (9,-9)), (Q = (1,5)), and (R=(16,1)).

Answer 1

Upon calculating the distances between the given points using the Euclidean distance formula, it's clear that the sum of any two pairwise distances does not equal the total distance, indicating that the points are not collinear. Thus, both options A and B are incorrect.

Step-by-step explanation:

To determine whether the given points are collinear, we need to check if the sum of the distances between any two pairs of points equals the distance between the first and last points. Let's calculate:

1. d(P, Q) where P = (1, 5) and Q = (16, 1).
2. d(Q, R) where Q = (16, 1) and R = (9, -9).
3. d(P, R) where P = (1, 5) and R = (9, -9).

The Euclidean distance formula is d(A, B) = √((x2 - x1)² + (y2 - y1)²), where A = (x1, y1) and B = (x2, y2).

Calculating the distances:

1. d(P, Q) = √((16 - 1)² + (1 - 5)²) = √(225 + 16) = √241
2. d(Q, R) = √((9 - 16)² + (-9 - 1)²) = √(49 + 100) = √149
3. d(P, R) = √((9 - 1)² + (-9 - 5)²) = √(64 + 196) = √260

Comparing the sums of the distances:

d(P, Q) + d(Q, R) = √241 + √149 differs from d(P, R) = √260, hence the given points are not collinear.

None of the provided answer options A or B are correct since both suggest that the points are collinear, which is refuted by our calculations.