{ "model": { "rank": 3586, "code": "thing-q158688", "model_id": "vr.tr.right-triangle", "name": "right triangle", "purpose": "Let an agent explain right triangles and their properties, relay the Pythagorean theorem, special triangles and trigonometric ratios from mathematical references, and describe applications in construction and surveying.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q158688/", "tier": 2, "score": 84, "payload": { "layer": "wikidata", "aliases": [ "90-30-60 set square", "isosceles right triangle", "30-60-90 triangle", "3-4-5 triangle", "Kepler triangle", "special right triangle", "Egyptian triangle" ], "aliasCount": 7, "merged": 7, "knownIn": 84, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q158688", "url": "https://www.wikidata.org/wiki/Q158688", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T02:44:35Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 3, "engine": "claude" }, "metaModel": { "id": "THING-Q158688", "registryId": "vr.tr.right-triangle", "name": "right triangle", "version": "0.2.0-wave.3", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q158688/", "model": { "registry_id": "vr.tr.right-triangle", "name": "right triangle", "purpose": "Let an agent explain right triangles and their properties, relay the Pythagorean theorem, special triangles and trigonometric ratios from mathematical references, and describe applications in construction and surveying.", "definition": "A triangle in which one angle is a right angle of 90 degrees, with the side opposite the right angle called the hypotenuse and the other two sides the legs, governed by the Pythagorean theorem, including special right triangles such as the isosceles 45-45-90 triangle, the 30-60-90 triangle used in set squares, the 3-4-5 triangle and other Pythagorean triples, and the Kepler triangle whose sides are in geometric progression with the golden ratio; right triangles are the basis of trigonometry and surveying.", "what_it_is_for": "Not applicable; a geometric figure.", "affordances": [ "explain properties", "relay the Pythagorean theorem and special triangles", "relay trigonometric ratios", "describe applications" ], "distinguishing_features": [ "Right angle", "Hypotenuse", "Pythagorean theorem", "Trigonometric basis" ], "appearance": "A triangle with one square corner.", "visual_identification": [ "Triangle with a 90 degree angle", "Isosceles right, 30-60-90, 3-4-5, Kepler and other special right triangles", "Obtuse and acute triangles have no right angle; set squares are tools shaped as right triangles" ], "physical_properties": [ { "quantity": "Pythagorean theorem", "typical_range": "a^2 + b^2 = c^2", "unit": "formula", "note": "" }, { "quantity": "45-45-90 side ratio", "typical_range": "1 : 1 : sqrt 2", "unit": "ratio", "note": "" }, { "quantity": "30-60-90 side ratio", "typical_range": "1 : sqrt 3 : 2", "unit": "ratio", "note": "" } ], "families_and_kinds": [ "isosceles right triangle", "30-60-90 triangle and set squares", "Pythagorean triple triangles such as 3-4-5", "Kepler triangle", "right triangles in coordinate geometry", "spherical right triangles" ], "related_models": [ { "relation": "is a kind of", "target": "non-equilateral triangle", "why": "in registry terms" }, { "relation": "is governed by", "target": "Pythagorean theorem", "why": "relating its sides" }, { "relation": "is the basis of", "target": "trigonometry", "why": "through side ratios" }, { "relation": "is embodied in", "target": "set square", "why": "the drawing tool" } ], "identifiers": [], "standards_and_regulation": [ "No regulation; standard geometric definitions", "Drawing instrument standards for set squares" ], "failure_modes_and_hazards": [ "Misidentifying the hypotenuse", "Applying the theorem to non-right triangles", "Confusing special triangle ratios" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What a right triangle is.", "rationale": "Mathematics.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition and parts.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a right triangle, and what are its legs, hypotenuse and angles?", "kind": "definition" }, { "text": "Is the question about right triangles in general, a special triangle or a set square?", "kind": "boundary" } ] } ] }, { "id": "special", "name": "Special", "description": "Special right triangles.", "findings": [ { "id": "special-finding", "name": "Special", "description": "Special.", "questions": [ { "text": "What are the 45-45-90, 30-60-90, 3-4-5 and Kepler triangles, and why are they useful?", "kind": "definition" }, { "text": "Which entry fits the specific triangle?", "kind": "action" } ] } ] } ] }, { "id": "compute", "name": "Compute", "description": "Computation.", "rationale": "Mathematics.", "layers": [ { "id": "pythagoras", "name": "Pythagoras", "description": "Pythagorean theorem.", "findings": [ { "id": "pythagoras-finding", "name": "Pythagoras", "description": "Pythagoras.", "questions": [ { "text": "How does the Pythagorean theorem work, and how is it proved?", "kind": "action" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "trigonometry", "name": "Trigonometry", "description": "Trigonometric ratios.", "findings": [ { "id": "trigonometry-finding", "name": "Trigonometry", "description": "Trigonometry.", "questions": [ { "text": "How do sine, cosine and tangent arise from right triangles, and how are unknown sides and angles found?", "kind": "action" }, { "text": "Which entry fits trigonometry?", "kind": "action" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Applications.", "rationale": "Practice.", "layers": [ { "id": "construction", "name": "Construction", "description": "Construction and surveying.", "findings": [ { "id": "construction-finding", "name": "Construction", "description": "Construction.", "questions": [ { "text": "How are 3-4-5 triangles and set squares used to lay out right angles in building and surveying?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] }, { "id": "science", "name": "Science", "description": "Science and engineering.", "findings": [ { "id": "science-finding", "name": "Science", "description": "Science.", "questions": [ { "text": "How do right triangles appear in vectors, navigation and physics?", "kind": "provenance" }, { "text": "Which entry fits vector decomposition?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "History and teaching.", "rationale": "Context.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did Babylonian, Egyptian, Greek, Indian and Chinese mathematicians use right triangles?", "kind": "provenance" }, { "text": "Which entry fits the history of geometry?", "kind": "action" } ] } ] }, { "id": "teaching", "name": "Teaching", "description": "Teaching.", "findings": [ { "id": "teaching-finding", "name": "Teaching", "description": "Teaching.", "questions": [ { "text": "How are right triangles taught, and what misconceptions arise?", "kind": "provenance" }, { "text": "Which entry fits mathematics education?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should special right triangle and Pythagorean triple be separate entries?", "How should mathematical references be linked?", "The registry entry has merged aliases naming specific triangles and a tool; should they be split off?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a right triangle, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain right triangles and their properties, relay the Pythagorean theorem, special triangles and trigonometric ratios from mathematical references, and describe applications in construction and surveying.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: 90-30-60 set square, isosceles right triangle, 30-60-90 triangle, 3-4-5 triangle, Kepler triangle, special right triangle, Egyptian triangle.", "7 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 84 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a right triangle, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "90-30-60 set square", "isosceles right triangle", "30-60-90 triangle", "3-4-5 triangle", "Kepler triangle", "special right triangle", "Egyptian triangle" ] } ], "standing": "Described in 84 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 7 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one right triangle from another, and a right triangle from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a right triangle, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a right triangle, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a right triangle is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a right triangle, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a right triangle, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a right triangle is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a right triangle sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a right triangle, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a right triangle, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a right triangle is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a right triangle from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a right triangle, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a right triangle, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a right triangle is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a right triangle passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a right triangle between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a right triangle, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a right triangle, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a right triangle is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a right triangle, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a right triangle is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a right triangle, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a right triangle, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a right triangle is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What right triangle is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of right triangle are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is right triangle operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of right triangle and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of right triangle?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as right triangle?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a right triangle came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a right triangle and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a right triangle, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a right triangle, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a right triangle is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a right triangle is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a right triangle, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a right triangle, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a right triangle is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }