# Question 3: Coordinate-Plane Distance

[Return to catalogue page 1](https://ocme.evemisslab.com/en/questions/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260912-geometry-foundations/#q-20260912-distance-coordinate-0003)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

P250-0003 · q-20260912-distance-coordinate-0003



## Original question and givens

In the Euclidean plane, P=(-9, -4) and Q=(-1, -19). Find the distance d(P,Q).


- G001 — In the Euclidean plane, P=(-9, -4) and Q=(-1, -19). 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.


- [mko-euclidean-length](https://ocme.evemisslab.com/en/mko/mko-euclidean-length/)





## Step-by-step derivation

S001 · impliesBefore

P=(-9,-4), Q=(-1,-19)

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 dependenciesRule: coordinate_difference · Uses: G001





S002 · calculationBefore

Δx := sub(-1, -9)

After

Δx = 8

Conditions: none

Subtract -9 from -1; keep parentheses around negative coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

Δy := sub(-19, -4)

After

Δy = -15

Conditions: none

The vertical difference is -19 minus -4.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

(Δx)² := square(8)

After

(Δx)² = 64

Conditions: none

Square the entire horizontal difference, not just the subtracted coordinate.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

(Δy)² := square(-15)

After

(Δy)² = 225

Conditions: none

The square is nonnegative even when the vertical difference is negative.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

d² := sum(64, 225)

After

d² = 289

Conditions: none

Add the two squares in the Euclidean distance formula.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

d²=289, d≥0

After

d=√(289)=17

Conditions: d≥0

Since 17²=289 and distance is nonnegative, choose the nonnegative root 17.

Step identifiers and dependenciesRule: nonnegative_square_root · Uses: G001, S006





S008 · calculationBefore

answer² := square(17)

After

answer² = 289

Conditions: none

Square the proposed distance to recover the original sum of squares.

Step identifiers and dependenciesRule: 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

17

The distance is 17.





## 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: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 00c72ac70baf02e0823e8a19b914029db3cc72a262f19c78dd4c0f4685d598d9
- candidate record SHA-256: 577c5f3d33165775644f52e91c2f09493936761d04d2149938cdf77bbbfc0e9d
- semantic review receipt SHA-256: ada91cd601dc368efd790cdd3aa80d0c7432d82b465fcdd1eef5148d3959755c
- question-first witness SHA-256: c36a153d3b48c84ba1bb7f6534cc6b835c243b74dd8080300b8026ab133fc6f6
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260912-distance-coordinate-0003.json)