Question 35541: Coordinate-Plane Distance

Original question and givens

In the Euclidean plane, P=(11, -16) and Q=(-7, 64). Find the distance d(P,Q).

• G001 — In the Euclidean plane, P=(11, -16) and Q=(-7, 64). 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 = [[11, -16], [-7, 64]]

derived = [orthonormal common coordinate frame]

Distance ORIGINAL coordinate difference i=0 := sub(-7, 11)

Distance ORIGINAL coordinate difference i=0 = -18

Distance square signed difference i=0 := mul(-18, -18)

Distance square signed difference i=0 = 324

Distance ORIGINAL coordinate difference i=1 := sub(64, -16)

Distance ORIGINAL coordinate difference i=1 = 80

Distance square signed difference i=1 := mul(80, 80)

Distance square signed difference i=1 = 6400

Distance sum BOTH coordinate squares := sum(324, 6400)

Distance sum BOTH coordinate squares = 6724

Distance principal nonnegative root squared back := mul(82, 82)

Distance principal nonnegative root squared back = 6724

source roles = [[-18, 80], 6724]

derived = [82]

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

82

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.