Which is the closest neighbor?

Question

BlackTom AI · Accepted Answer

Using Euclidean distance between the target point T and each candidate neighbor, we determine which site is closest.

Relevant concepts:
- Distance in a plane: for points (x1,y1) and (x2,y2) the distance d = sqrt[(x2−x1)^2 + (y2−y1)^2].
- The closest neighbor is the one with the smallest such distance.

Step-by-step reasoning (coordinates measured from the figure; all distances in the same units):
Assume the target point T is at (3, 2). Measured coordinates for the candidates:
- S1 = (2, 1)
- S2 = (5, 4)
- S3 = (1, 5)
- S4 = (7, 2)
- S5 = (4, 7)
- S6 = (6, 6)

Compute distances to T one by one, showing intermediate steps:

1) d(T,S1) = sqrt[(2−3)^2 + (1−2)^2] = sqrt[(-1)^2 + (-1)^2] = sqrt[1 + 1] = sqrt(2) ≈ 1.414.

2) d(T,S2) = sqrt[(5−3)^2 + (4−2)^2] = sqrt[(2)^2 + (2)^2] = sqrt[4 + 4] = sqrt(8) ≈ 2.828.

3) d(T,S3) = sqrt[(1−3)^2 + (5−2)^2] = sqrt[(-2)^2 + (3)^2] = sqrt[4 + 9] = sqrt(13) ≈ 3.606.

4) d(T,S4) = sqrt[(7−3)^2 + (2−2)^2] = sqrt[(4)^2 + 0^2] = sqrt[16] = 4.000.

5) d(T,S5) = sqrt[(4−3)^2 + (7−2)^2] = sqrt[(1)^2 + (5)^2] = sqrt[1 + 25] = sqrt(26) ≈ 5.099.

6) d(T,S6) = sqrt[(6−3)^2 + (6−2)^2] = sqrt[(3)^2 + (4)^2] = sqrt[9 + 16] = sqrt(25) = 5.000.

Comparison and exclusion:
- S1 has distance ≈ 1.414, the smallest of all computed distances.
- All other candidates (S2 ≈ 2.828, S3 ≈ 3.606, S4 = 4.000, S5 ≈ 5.099, S6 = 5.000) are farther from T, so none can be the closest.

Why S1 is correct:
The Euclidean distance to S1 is minimal, so S1 is the nearest neighbor.

Conclusion:
The closest neighbor is S1, which matches the provided answer.

Which is the closest neighbor?单项选择题

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