# Question 29745: Counterclockwise Quarter-Turns about the Origin [Return to catalogue page 149](https://ocme.evemisslab.com/en/questions/page/149/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261006-advance-20261007-r07-geometric-transformations-foundations/#q-20261006-advance-20261007-r07-geometric-transformations-0045) 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. R0058-29745 · q-20261006-advance-20261007-r07-geometric-transformations-0045 ## Original question and givens Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(7, 20) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates. - G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(7, 20) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates. ## 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-quarter-turn-point, quarter_turn_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 CCW turn1 negated old y := sub(0, 20) After Transform ORIGINAL CCW turn1 negated old y = -20 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 · impliesBefore source roles = [[7, 20], 3, 0] After derived = [[-20, 7]] Conditions: none Each step is a90-degree COUNTERCLOCKWISE rotation about the ORIGIN,current(x,y)->(-y,x).Do not change center/direction.The source permits ONLY0..3,not modulo repair of negative counts or4. Step identifiers and dependenciesRule: Transform_EACH_ORIGIN_counterclockwise_quarter_turn · Uses: G001, S002 S004 · calculationBefore Transform ORIGINAL CCW turn2 negated old y := sub(0, 7) After Transform ORIGINAL CCW turn2 negated old y = -7 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, S003 S005 · impliesBefore source roles = [[7, 20], 3, 1] After derived = [[-7, -20]] Conditions: none Each step is a90-degree COUNTERCLOCKWISE rotation about the ORIGIN,current(x,y)->(-y,x).Do not change center/direction.The source permits ONLY0..3,not modulo repair of negative counts or4. Step identifiers and dependenciesRule: Transform_EACH_ORIGIN_counterclockwise_quarter_turn · Uses: G001, S004 S006 · calculationBefore Transform ORIGINAL CCW turn3 negated old y := sub(0, -20) After Transform ORIGINAL CCW turn3 negated old y = 20 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, S005 S007 · impliesBefore source roles = [[7, 20], 3, 2] After derived = [[20, -7]] Conditions: none Each step is a90-degree COUNTERCLOCKWISE rotation about the ORIGIN,current(x,y)->(-y,x).Do not change center/direction.The source permits ONLY0..3,not modulo repair of negative counts or4. Step identifiers and dependenciesRule: Transform_EACH_ORIGIN_counterclockwise_quarter_turn · Uses: G001, S006 S008 · impliesBefore source roles = [[7, 20], 3] After derived = [[20, -7]] Conditions: none At n=0 retain the original point;otherwise use exactly the stated updates.This image calculation is not a new formal certificate for a whole-plane theorem. Step identifiers and dependenciesRule: Transform_actual_declared_turn_count_including_zero · Uses: G001, S007 ## Reverse checks - S008 · 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. - S007 · 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. - S006 · premise_and_result_audit — Recompute Transform ORIGINAL CCW turn3 negated old y with exact rational arithmetic, not rounded decimals. - S005 · 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. - S004 · premise_and_result_audit — Recompute Transform ORIGINAL CCW turn2 negated old y with exact rational arithmetic, not rounded decimals. - S003 · 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. - S002 · premise_and_result_audit — Recompute Transform ORIGINAL CCW turn1 negated old y 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 (20, -7) 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: 79d3f1c030c164a15258d4e5b12f2f8fb0d39be218da012a033c9d146540a330 - candidate record SHA-256: 4a3b4ac7f120ab236933fb48881f71a832ce6dafe48a11782000482b494d0f46 - 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-0045.json)