Question 35542: Coordinate-Plane Distance

Original question and givens

In the Euclidean plane, P=(18, -5) and Q=(42, -75). Find the distance d(P,Q).

• G001 — In the Euclidean plane, P=(18, -5) and Q=(42, -75). Find the distance d(P,Q).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-euclidean-length

Step-by-step derivation

source roles = [qf-distance-coordinate, [expected_distance, p, q]]

derived = [distance, null, null]

source roles = [[18, -5], [42, -75]]

derived = [orthonormal common coordinate frame]

Distance ORIGINAL coordinate difference i=0 := sub(42, 18)

Distance ORIGINAL coordinate difference i=0 = 24

Distance square signed difference i=0 := mul(24, 24)

Distance square signed difference i=0 = 576

Distance ORIGINAL coordinate difference i=1 := sub(-75, -5)

Distance ORIGINAL coordinate difference i=1 = -70

Distance square signed difference i=1 := mul(-70, -70)

Distance square signed difference i=1 = 4900

Distance sum BOTH coordinate squares := sum(576, 4900)

Distance sum BOTH coordinate squares = 5476

Distance principal nonnegative root squared back := mul(74, 74)

Distance principal nonnegative root squared back = 5476

source roles = [[24, -70], 5476]

derived = [74]

Reverse checks

• S009 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
• S008 · premise_and_result_audit — Recompute Distance principal nonnegative root squared back with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Distance sum BOTH coordinate squares with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Distance square signed difference i=1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Distance ORIGINAL coordinate difference i=1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Distance square signed difference i=0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Distance ORIGINAL coordinate difference i=0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
• S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.

Result

74

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.