S001 · implies
Before
P=(-21,-2), Q=(-32,-62)
After
Δx=Q_x−P_x; Δy=Q_y−P_y
Conditions: none
Use the same direction Q−P for both coordinate differences.
Step identifiers and dependencies
Rule: coordinate_difference · Uses: G001
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Question 8: Coordinate-Plane Distance
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← Question derivation index · Original question batch
P250-0008 · q-20260912-distance-coordinate-0008
Original question and givens
In the Euclidean plane, P=(-21, -2) and Q=(-32, -62). Find the distance d(P,Q).
G001— In the Euclidean plane, P=(-21, -2) and Q=(-32, -62). Find the distance d(P,Q).
Definition and scope
Distance uses the nonnegative principal square root, following existing source assumption a3. Coincident points have zero distance; the selected pairs are distinct and have positive distances.
Step-by-step derivation
S002 · calculation
Before
Δx := sub(-32, -21)
After
Δx = -11
Conditions: none
Subtract -21 from -32; keep parentheses around negative coordinates.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Δy := sub(-62, -2)
After
Δy = -60
Conditions: none
The vertical difference is -62 minus -2.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
(Δx)² := square(-11)
After
(Δx)² = 121
Conditions: none
Square the entire horizontal difference, not just the subtracted coordinate.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
(Δy)² := square(-60)
After
(Δy)² = 3600
Conditions: none
The square is nonnegative even when the vertical difference is negative.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
d² := sum(121, 3600)
After
d² = 3721
Conditions: none
Add the two squares in the Euclidean distance formula.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
d²=3721, d≥0
After
d=√(3721)=61
Conditions: d≥0
Since 61²=3721 and distance is nonnegative, choose the nonnegative root 61.
Step identifiers and dependencies
Rule: nonnegative_square_root · Uses: G001, S006
S008 · calculation
Before
answer² := square(61)
After
answer² = 3721
Conditions: none
Square the proposed distance to recover the original sum of squares.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
Reverse checks
S008· premise_and_result_audit — Recompute answer² with exact rational arithmetic, not rounded decimals.S007· premise_and_result_audit — The negative root is not a distance, and squared distance is not distance itself.S006· premise_and_result_audit — Recompute d² with exact rational arithmetic, not rounded decimals.S005· premise_and_result_audit — Recompute (Δy)² with exact rational arithmetic, not rounded decimals.S004· premise_and_result_audit — Recompute (Δx)² with exact rational arithmetic, not rounded decimals.S003· premise_and_result_audit — Recompute Δy with exact rational arithmetic, not rounded decimals.S002· premise_and_result_audit — Recompute Δx with exact rational arithmetic, not rounded decimals.S001· premise_and_result_audit — Swapping the points changes both signs but not the sum of squares.
Result
61
The distance is 61.
This question review and source binding
Review date: 2026-10-24
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
Source hashes and review receipts
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 7ecff48f00b86f1ae703fc8ac6cb3f1883944ce2299b5e5248f089ad7f771ba2candidate record SHA-256: df10e16f3ff0f318a48a5d1a7157f1d797966adc7bc892faef3d5a779bb6db58semantic review receipt SHA-256: ada91cd601dc368efd790cdd3aa80d0c7432d82b465fcdd1eef5148d3959755cquestion-first witness SHA-256: c36a153d3b48c84ba1bb7f6534cc6b835c243b74dd8080300b8026ab133fc6f6Lean: NOT_RUN · human review: NOT_PERFORMED