# Question 29758: Exact Image Point under Translation

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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.

R0058-29758 · q-20261006-advance-20261007-r07-geometric-transformations-0008



## Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-5, -6) and translation vector t=[0, 2]. Find the coordinates of P+t.


- G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-5, -6) and translation vector t=[0, 2]. Find the coordinates of P+t.





## 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-geometric-transformations](https://ocme.evemisslab.com/en/mko/mko-geometric-transformations/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-transform-translation-point, translation_point]

After

derived = [[image_POINT_string]]

Conditions: none

Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.

Step identifiers and dependenciesRule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001





S002 · calculationBefore

Transform ORIGINAL translation coordinate0 := sum(-5, 0)

After

Transform ORIGINAL translation coordinate0 = -5

Conditions: none

Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.

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





S003 · calculationBefore

Transform ORIGINAL translation coordinate1 := sum(-6, 2)

After

Transform ORIGINAL translation coordinate1 = -4

Conditions: none

Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.

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





S004 · impliesBefore

source roles = [[-5, -6], [0, 2]]

After

derived = [[-5, -4]]

Conditions: none

In the SAME origin/basis,add each original translation component to obtain the ACTUAL image.Returning only v is not P+t.Derived sums may exceed SAFE while raw inputs remain SAFE.

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









## Reverse checks


- S004 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
- S003 · premise_and_result_audit — Recompute Transform ORIGINAL translation coordinate1 with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Transform ORIGINAL translation coordinate0 with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.





## Result

(-5, -4)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: aaa730a505fb3ed46ae90f8eb3a47e54e03888c0d2166e21478711a0ed584715
- candidate record SHA-256: 684f2f85de62554b358c5f5980c8c012551d2c10a4778d5d4072daa9f74887ba
- semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df
- question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261006-advance-20261007-r07-geometric-transformations-0008.json)