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Culture","lang",{"id":564,"unique_code":565,"title":566,"icon":8,"intro":567,"cate":483,"detail":568,"create_time":569,"update_time":570,"price":571,"view_count":572,"download_count":573,"unit":573,"amount":574,"is_create_order":573,"enterprise_id":488,"seo_keywords":575,"seo_description":576,"data_format":577,"file_size":578,"apply_scene":579,"training_task":580,"version":581,"dataset_type_id":510,"dataset_scene_id":505,"sample_file":582,"banner":8,"unit_name":8,"cate_name":449,"icon_url":8,"banner_url":8,"sample_file_url":8,"data_dict":583,"qa_list":626,"sample_metadatas":643,"text_sample_content":683,"enterprise_name":684,"enterprise_logo":685,"data_distribution":686},1218,"978e10b8a93cc6c184790b670a704d3a","SAT Math Concepts and Formula Evidence Retrieval Dataset","Connect SAT math questions to traceable concept, formula, and theorem evidence to build grounded solution retrieval workflows.","\u003Cp>Designed for SAT math questions, this dataset pairs original question text and structured conditions with relevant concept definitions, formulas or theorems, applicability conditions, solution links, and source materials. Its JSON records include consolidated retrieval text and an evidence quality score ranging from 0 to 1, supporting workflows that retrieve reliable mathematical grounds for each question. Typical uses include RAG knowledge retrieval, evidence-backed explanations, and retrieval system development and evaluation.\u003C\u002Fp>","2026-10-14 18:09:06","2026-10-19 17:20:14","0.00",22,0,5000,"SAT math dataset, SAT math questions, math formula retrieval, mathematical evidence, RAG training data, evidence retrieval, structured math problems, knowledge grounding","A JSON dataset linking SAT math questions to structured conditions, sourced formulas and theorems, and solution evidence for grounded RAG retrieval and tutoring","JSON","0.5GB","SAT math concept retrieval, mathematical formula lookup, theorem-based problem solving","information retrieval, question answering, evidence retrieval","V1.0","rivo\u002Fsample-packages\u002F0aaa3668-775c-4b95-9562-5aabc1799946\u002F20261014.zip",[584,589,594,599,605,610,616,621],{"id":573,"name":585,"field":586,"intro":587,"type":588},"File Name","file_name","File name","string",{"id":590,"name":591,"field":592,"intro":593,"type":588},8863,"Question Text","question_text","The complete original SAT math question, including its prompt, requested task, and mathematical expressions.",{"id":595,"name":596,"field":597,"intro":598,"type":588},8864,"Retrieval Chain Text","retrieval_text","Evidence text combining question conditions, concepts, formulas or theorems, applicability conditions, and solution links for RAG retrieval and generation.",{"id":600,"name":601,"field":602,"intro":603,"type":604},8865,"Evidence Quality Score","evidence_quality","A composite score from 0 to 1 assessing mathematical evidence accuracy, source traceability, and relevance to solving the question.","number",{"id":606,"name":607,"field":608,"intro":609,"type":588},8866,"Source Knowledge Materials","source_knowledge","Original materials used to extract or verify mathematical concepts, definitions, formulas, theorems, and their sources. May include textbooks, official resources, or curated knowledge excerpts.",{"id":611,"name":612,"field":613,"intro":614,"type":615},8867,"Solution Relationship Annotations","solution_relation","Annotations describing how the question conditions and mathematical evidence correspond to the solution process.","object",{"id":617,"name":618,"field":619,"intro":620,"type":615},8868,"Mathematical Knowledge Evidence","knowledge_evidence","Concept definitions, formulas or theorems, applicability conditions, and reliable source evidence relevant to solving the question.",{"id":622,"name":623,"field":624,"intro":625,"type":615},8869,"Structured Question Conditions","structured_conditions","Structured information extracted from the question text, such as question type, given conditions, objective, and answer choices.",[627,631,635,639],{"id":628,"question":629,"answer":630},6323,"What does the dataset include?","Records are organized around SAT math questions and include question text, structured conditions, source knowledge materials, mathematical evidence, solution links, retrieval text, and an evidence quality score.",{"id":632,"question":633,"answer":634},6324,"What does the evidence quality score indicate?","The score ranges from 0 to 1 and reflects evidence accuracy, source traceability, and relevance to solving the question. It can help with filtering and evaluation, but important instructional or assessment use should still involve checking the original sources and evidence.",{"id":636,"question":637,"answer":638},6325,"What are the intended use cases?","The dataset supports SAT math RAG retrieval and generation, knowledge grounding, evidence-backed solution explanations, and retrieval workflow development and evaluation. Its structured conditions and retrieval text can also support matching questions to concepts, formulas, or theorems.",{"id":640,"question":641,"answer":642},6326,"How can I access the data, and what are the licensing terms?","The data is organized in JSON; refer to the product page or provider's license for specific download methods, availability, and usage terms. Any textbooks, official materials, or other referenced sources remain subject to their applicable copyright and usage requirements.",[644,659,671],{"id":645,"relate_table":646,"relate_id":647,"file_size":573,"file_name":648,"path":8,"duration":573,"cate":483,"file_format":649,"audio_rate":8,"audio_channel":522,"audio_bit_depth":522,"quality":8,"annotation":650,"cate_name":651,"question_text":652,"retrieval_text":653,"evidence_quality":654,"source_knowledge":655,"solution_relation":656,"knowledge_evidence":657,"structured_conditions":658},15054,"union_cubo",149636082,"样例_3.json","json","{\"question_text\": \"In a circle, AB is a diameter and C is a point on the circle. If AC=9 and BC=12, what is the area of the circle?\", \"retrieval_text\": \"In the circle, AB is a diameter, C lies on the circle, AC=9, and BC=12. By the theorem that an inscribed angle subtended by a diameter is a right angle, triangle ABC is a right triangle with AB as its hypotenuse. By the Pythagorean theorem, AB²=9²+12²=225, so the diameter AB=15 and the radius r=15\u002F2. The area of the circle is A=πr²=π(15\u002F2)²=225π\u002F4. This problem is a combined application of the inscribed angle theorem, the Pythagorean theorem, and the circle area formula.\", \"evidence_quality\": \"0.93\", \"source_knowledge\": \"Circle and right triangle knowledge material: An inscribed angle with a diameter of the circle as one side and another point on the circle as its vertex is a right angle. Therefore, when AB is a diameter and C lies on the circle, triangle ABC is right-angled at C. The diameter AB is the hypotenuse of this right triangle and can be found using the Pythagorean theorem; then divide the diameter by 2 to get the radius, and finally use the circle area formula.\", \"solution_relation\": \"{\\\"relation_type\\\": \\\"Combined application\\\", \\\"linked_concepts\\\": [\\\"An inscribed angle subtended by a diameter is a right angle\\\", \\\"Pythagorean theorem\\\", \\\"Radius of a circle\\\", \\\"Area of a circle\\\"], \\\"reasoning_link\\\": \\\"Because AB is a diameter and C lies on the circle, ∠ACB is a right angle, AC and BC are the two legs, and AB is the hypotenuse. First use the Pythagorean theorem to find the diameter, then convert it to the radius and calculate the area of the circle.\\\", \\\"answer_derivation\\\": \\\"AB²=9²+12²=81+144=225, so AB=15. The radius r=15\u002F2. The area of the circle is A=π(15\u002F2)²=225π\u002F4, so the correct answer is 225π\u002F4.\\\", \\\"limitation_notes\\\": \\\"Do not directly assume the triangle is right-angled merely from AC=9 and BC=12; the right angle comes from AB being a diameter and C lying on the circle. Another common mistake is treating 15 as the radius, which incorrectly makes the area four times too large.\\\"}\", \"knowledge_evidence\": \"{\\\"concept_name\\\": \\\"Inscribed angle theorem, Pythagorean theorem, and area of a circle\\\", \\\"concept_definition\\\": \\\"If all three vertices of a triangle lie on the same circle and one side is a diameter of the circle, then the angle subtended by that diameter is a right angle. Therefore, the diameter is the hypotenuse of that right triangle. The side lengths of a right triangle satisfy the Pythagorean theorem, and the area of a circle equals π times the square of the radius.\\\", \\\"formula_or_theorem\\\": \\\"∠ACB=90°; AB²=AC²+BC²; A=πr², and r=AB\u002F2.\\\", \\\"applicability_conditions\\\": [\\\"AB must actually be a diameter of the circle, and A, B, and C must all lie on the same circle.\\\", \\\"The Pythagorean theorem can only be used for right triangles.\\\", \\\"The radius of a circle equals half the diameter; the diameter cannot be substituted directly into the circle area formula.\\\"], \\\"source_basis\\\": \\\"CK-12 Geometry FlexBook 2.0, chapters related to Right Triangles and Circles; the source notes and evidence excerpts in this sample are paraphrased in Chinese, not verbatim quotations.\\\", \\\"evidence_excerpt\\\": \\\"Chinese paraphrase excerpt: An inscribed angle subtended by a diameter is a right angle; in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs; the area of a circle is given by πr².\\\"}\", \"structured_conditions\": \"{\\\"question_type\\\": \\\"Geometry: Circles, right triangles, and the Pythagorean theorem\\\", \\\"given_conditions\\\": [\\\"AB is a diameter of the circle.\\\", \\\"Point C lies on the circle.\\\", \\\"AC=9.\\\", \\\"BC=12.\\\", \\\"The formula for the area of a circle is A=πr².\\\"], \\\"target_quantity\\\": \\\"Find the area of the circle.\\\", \\\"answer_choices\\\": [\\\"225π\u002F16\\\", \\\"225π\u002F4\\\", \\\"225π\u002F2\\\", \\\"450π\\\"]}\"}","文本","In a circle, AB is a diameter and C is a point on the circle. If AC=9 and BC=12, what is the area of the circle?","In the circle, AB is a diameter, C lies on the circle, AC=9, and BC=12. By the theorem that an inscribed angle subtended by a diameter is a right angle, triangle ABC is a right triangle with AB as its hypotenuse. By the Pythagorean theorem, AB²=9²+12²=225, so the diameter AB=15 and the radius r=15\u002F2. The area of the circle is A=πr²=π(15\u002F2)²=225π\u002F4. This problem is a combined application of the inscribed angle theorem, the Pythagorean theorem, and the circle area formula.","0.93","Circle and right triangle knowledge material: An inscribed angle with a diameter of the circle as one side and another point on the circle as its vertex is a right angle. Therefore, when AB is a diameter and C lies on the circle, triangle ABC is right-angled at C. The diameter AB is the hypotenuse of this right triangle and can be found using the Pythagorean theorem; then divide the diameter by 2 to get the radius, and finally use the circle area formula.","{\"relation_type\": \"Combined application\", \"linked_concepts\": [\"An inscribed angle subtended by a diameter is a right angle\", \"Pythagorean theorem\", \"Radius of a circle\", \"Area of a circle\"], \"reasoning_link\": \"Because AB is a diameter and C lies on the circle, ∠ACB is a right angle, AC and BC are the two legs, and AB is the hypotenuse. First use the Pythagorean theorem to find the diameter, then convert it to the radius and calculate the area of the circle.\", \"answer_derivation\": \"AB²=9²+12²=81+144=225, so AB=15. The radius r=15\u002F2. The area of the circle is A=π(15\u002F2)²=225π\u002F4, so the correct answer is 225π\u002F4.\", \"limitation_notes\": \"Do not directly assume the triangle is right-angled merely from AC=9 and BC=12; the right angle comes from AB being a diameter and C lying on the circle. Another common mistake is treating 15 as the radius, which incorrectly makes the area four times too large.\"}","{\"concept_name\": \"Inscribed angle theorem, Pythagorean theorem, and area of a circle\", \"concept_definition\": \"If all three vertices of a triangle lie on the same circle and one side is a diameter of the circle, then the angle subtended by that diameter is a right angle. Therefore, the diameter is the hypotenuse of that right triangle. The side lengths of a right triangle satisfy the Pythagorean theorem, and the area of a circle equals π times the square of the radius.\", \"formula_or_theorem\": \"∠ACB=90°; AB²=AC²+BC²; A=πr², and r=AB\u002F2.\", \"applicability_conditions\": [\"AB must actually be a diameter of the circle, and A, B, and C must all lie on the same circle.\", \"The Pythagorean theorem can only be used for right triangles.\", \"The radius of a circle equals half the diameter; the diameter cannot be substituted directly into the circle area formula.\"], \"source_basis\": \"CK-12 Geometry FlexBook 2.0, chapters related to Right Triangles and Circles; the source notes and evidence excerpts in this sample are paraphrased in Chinese, not verbatim quotations.\", \"evidence_excerpt\": \"Chinese paraphrase excerpt: An inscribed angle subtended by a diameter is a right angle; in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs; the area of a circle is given by πr².\"}","{\"question_type\": \"Geometry: Circles, right triangles, and the Pythagorean theorem\", \"given_conditions\": [\"AB is a diameter of the circle.\", \"Point C lies on the circle.\", \"AC=9.\", \"BC=12.\", \"The formula for the area of a circle is A=πr².\"], \"target_quantity\": \"Find the area of the circle.\", \"answer_choices\": [\"225π\u002F16\", \"225π\u002F4\", \"225π\u002F2\", \"450π\"]}",{"id":660,"relate_table":646,"relate_id":661,"file_size":573,"file_name":662,"path":8,"duration":573,"cate":483,"file_format":649,"audio_rate":8,"audio_channel":522,"audio_bit_depth":522,"quality":8,"annotation":663,"cate_name":651,"question_text":664,"retrieval_text":665,"evidence_quality":666,"source_knowledge":667,"solution_relation":668,"knowledge_evidence":669,"structured_conditions":670},15053,2088014468,"样例_2.json","{\"question_text\": \"A town has a population of 5,000. If its population increases by 6 percent each year, what will the population be after 2 years, assuming the growth rate remains constant?\", \"retrieval_text\": \"The problem gives an initial population of 5,000, a fixed annual growth rate of 6%, and a time of 2 years. Fixed percentage growth corresponds to the exponential model P=P₀(1+r)^t. Substituting P₀=5000, r=0.06, t=2 gives P=5000(1.06)^2=5618, so the correct answer is 5,618. The key applicability condition is that the growth proportion is the same each year; if a fixed number of people is added, the model would change to linear growth.\", \"evidence_quality\": \"0.94\", \"source_knowledge\": \"Exponential growth knowledge material: When a quantity grows by a fixed percentage, it is multiplied by the same growth factor each time period. If the initial quantity is P₀ and the periodic growth rate is r, then the quantity after t periods is P=P₀(1+r)^t. Here the growth rate must be expressed as a decimal.\", \"solution_relation\": \"{\\\"relation_type\\\": \\\"Direct application\\\", \\\"linked_concepts\\\": [\\\"Percentage growth\\\", \\\"Growth factor\\\", \\\"Exponential function\\\", \\\"Exponential growth model\\\"], \\\"reasoning_link\\\": \\\"The problem states that the population increases by a fixed 6% each year, so the amount added each year cannot be fixed at 300; instead, the current population must be multiplied by 1.06 each year. The initial population corresponds to P₀, and 2 years corresponds to t=2.\\\", \\\"answer_derivation\\\": \\\"Substitute into the exponential growth formula: P=5000(1+0.06)^2=5000(1.06)^2=5000×1.1236=5618. Therefore the model's projected population after 2 years is 5,618, corresponding to choice 5,618.\\\", \\\"limitation_notes\\\": \\\"Common errors include writing 6% as 6, calculating only one growth step, or fixing the annual increase at 6% of the initial population. If the problem states that a fixed number of people is added each year, a linear model should be used instead of an exponential model.\\\"}\", \"knowledge_evidence\": \"{\\\"concept_name\\\": \\\"Exponential growth model\\\", \\\"concept_definition\\\": \\\"If a quantity grows by a fixed percentage in every equal time period, it exhibits exponential growth. The growth factor per period is 1+r, where r is the positive growth rate expressed as a decimal.\\\", \\\"formula_or_theorem\\\": \\\"P=P₀(1+r)^t, where P₀ is the initial quantity, r is the growth rate per period, and t is the number of periods.\\\", \\\"applicability_conditions\\\": [\\\"The percentage growth rate per period remains constant.\\\", \\\"The growth rate r should be substituted into the formula as a decimal; for example, 6% should be written as 0.06.\\\", \\\"t represents a whole number of identical time periods.\\\", \\\"The model gives a projected value calculated according to the model; it does not mean the actual population must be an integer or that actual growth is completely stable.\\\"], \\\"source_basis\\\": \\\"OpenStax, Algebra and Trigonometry 2e, content on exponential growth and decay models in the Exponential and Logarithmic Functions chapter; the source notes and evidence excerpts in this sample are paraphrased in Chinese, not verbatim quotations.\\\", \\\"evidence_excerpt\\\": \\\"Chinese paraphrase excerpt: The exponential growth model multiplies the initial quantity by the growth factor raised to the t-th power; for a case with growth rate r per period, the growth factor is 1+r.\\\"}\", \"structured_conditions\": \"{\\\"question_type\\\": \\\"Problem solving and data analysis: Exponential growth model\\\", \\\"given_conditions\\\": [\\\"Initial population is 5,000.\\\", \\\"Annual growth rate is 6%, i.e., r=0.06.\\\", \\\"The growth rate remains constant each year.\\\", \\\"The time period for calculation is 2 years.\\\"], \\\"target_quantity\\\": \\\"Find the projected population after 2 years.\\\", \\\"answer_choices\\\": [\\\"5,300\\\", \\\"5,600\\\", \\\"5,618\\\", \\\"5,636\\\"]}\"}","A town has a population of 5,000. If its population increases by 6 percent each year, what will the population be after 2 years, assuming the growth rate remains constant?","The problem gives an initial population of 5,000, a fixed annual growth rate of 6%, and a time of 2 years. Fixed percentage growth corresponds to the exponential model P=P₀(1+r)^t. Substituting P₀=5000, r=0.06, t=2 gives P=5000(1.06)^2=5618, so the correct answer is 5,618. The key applicability condition is that the growth proportion is the same each year; if a fixed number of people is added, the model would change to linear growth.","0.94","Exponential growth knowledge material: When a quantity grows by a fixed percentage, it is multiplied by the same growth factor each time period. If the initial quantity is P₀ and the periodic growth rate is r, then the quantity after t periods is P=P₀(1+r)^t. Here the growth rate must be expressed as a decimal.","{\"relation_type\": \"Direct application\", \"linked_concepts\": [\"Percentage growth\", \"Growth factor\", \"Exponential function\", \"Exponential growth model\"], \"reasoning_link\": \"The problem states that the population increases by a fixed 6% each year, so the amount added each year cannot be fixed at 300; instead, the current population must be multiplied by 1.06 each year. The initial population corresponds to P₀, and 2 years corresponds to t=2.\", \"answer_derivation\": \"Substitute into the exponential growth formula: P=5000(1+0.06)^2=5000(1.06)^2=5000×1.1236=5618. Therefore the model's projected population after 2 years is 5,618, corresponding to choice 5,618.\", \"limitation_notes\": \"Common errors include writing 6% as 6, calculating only one growth step, or fixing the annual increase at 6% of the initial population. If the problem states that a fixed number of people is added each year, a linear model should be used instead of an exponential model.\"}","{\"concept_name\": \"Exponential growth model\", \"concept_definition\": \"If a quantity grows by a fixed percentage in every equal time period, it exhibits exponential growth. The growth factor per period is 1+r, where r is the positive growth rate expressed as a decimal.\", \"formula_or_theorem\": \"P=P₀(1+r)^t, where P₀ is the initial quantity, r is the growth rate per period, and t is the number of periods.\", \"applicability_conditions\": [\"The percentage growth rate per period remains constant.\", \"The growth rate r should be substituted into the formula as a decimal; for example, 6% should be written as 0.06.\", \"t represents a whole number of identical time periods.\", \"The model gives a projected value calculated according to the model; it does not mean the actual population must be an integer or that actual growth is completely stable.\"], \"source_basis\": \"OpenStax, Algebra and Trigonometry 2e, content on exponential growth and decay models in the Exponential and Logarithmic Functions chapter; the source notes and evidence excerpts in this sample are paraphrased in Chinese, not verbatim quotations.\", \"evidence_excerpt\": \"Chinese paraphrase excerpt: The exponential growth model multiplies the initial quantity by the growth factor raised to the t-th power; for a case with growth rate r per period, the growth factor is 1+r.\"}","{\"question_type\": \"Problem solving and data analysis: Exponential growth model\", \"given_conditions\": [\"Initial population is 5,000.\", \"Annual growth rate is 6%, i.e., r=0.06.\", \"The growth rate remains constant each year.\", \"The time period for calculation is 2 years.\"], \"target_quantity\": \"Find the projected population after 2 years.\", \"answer_choices\": [\"5,300\", \"5,600\", \"5,618\", \"5,636\"]}",{"id":672,"relate_table":646,"relate_id":673,"file_size":573,"file_name":674,"path":8,"duration":573,"cate":483,"file_format":649,"audio_rate":8,"audio_channel":522,"audio_bit_depth":522,"quality":8,"annotation":675,"cate_name":651,"question_text":676,"retrieval_text":677,"evidence_quality":678,"source_knowledge":679,"solution_relation":680,"knowledge_evidence":681,"structured_conditions":682},15052,1717160961,"样例_1.json","{\"question_text\": \"A line passes through the points (2, 5) and (8, 17). What is the x-coordinate of the point where the line intersects the x-axis?\", \"retrieval_text\": \"The problem states that a line passes through (2,5) and (8,17), and asks for its x-intercept. Using the slope formula m=(17−5)\u002F(8−2)=2, the point-slope form gives y−5=2(x−2), i.e., y=2x+1. Points on the x-axis satisfy y=0, so 0=2x+1, x=-1\u002F2. This problem directly applies two-point slope, point-slope form, and the definition of intercept; the applicable preconditions are that the two points have different x-coordinates and do lie on the same line.\", \"evidence_quality\": \"0.95\", \"source_knowledge\": \"Linear function knowledge material: The slope of a non-vertical line equals the ratio of the difference in y-coordinates to the difference in x-coordinates of two points. A line can usually be written in slope-intercept form y=mx+b, where m is the slope and b is the y-intercept. The intersection with the x-axis satisfies y=0.\", \"solution_relation\": \"{\\\"relation_type\\\": \\\"Direct application\\\", \\\"linked_concepts\\\": [\\\"Two-point slope\\\", \\\"Point-slope form\\\", \\\"Slope-intercept form\\\", \\\"x-intercept\\\"], \\\"reasoning_link\\\": \\\"The two points given in the problem directly provide the coordinates needed to calculate the slope. Substituting (2,5) and (8,17) into the slope formula gives m=2, then the point-slope form is used to construct the line equation, and finally the condition y=0 on the x-axis is used to find the intercept.\\\", \\\"answer_derivation\\\": \\\"m=(17−5)\u002F(8−2)=2. Using point-slope form: y−5=2(x−2), which simplifies to y=2x+1. Setting y=0 gives 0=2x+1, so x=-1\u002F2, corresponding to choice -1\u002F2.\\\", \\\"limitation_notes\\\": \\\"Do not mistake the x-intercept for the y-intercept; the x-intercept requires y=0, while the y-intercept requires x=0. Also note that the order of coordinate differences in the slope calculation must be consistent.\\\"}\", \"knowledge_evidence\": \"{\\\"concept_name\\\": \\\"Slope, slope-intercept form, and x-intercept\\\", \\\"concept_definition\\\": \\\"Given two points with different x-coordinates, the slope of the line is m=(y₂−y₁)\u002F(x₂−x₁). The slope-intercept form of the line is y=mx+b; setting y=0 gives the x-coordinate of its intersection with the x-axis.\\\", \\\"formula_or_theorem\\\": \\\"m=(y₂−y₁)\u002F(x₂−x₁); y−y₁=m(x−x₁); the x-intercept is found by setting y=0.\\\", \\\"applicability_conditions\\\": [\\\"The x-coordinates of the two points must be different; otherwise the denominator in the slope formula is zero.\\\", \\\"The two points must determine the same line.\\\", \\\"When finding the x-intercept, y must be set to 0.\\\"], \\\"source_basis\\\": \\\"OpenStax, Algebra and Trigonometry 2e, content on slope and equations of lines in the Linear Functions chapter; the source notes and evidence excerpts in this sample are paraphrased in Chinese, not verbatim quotations.\\\", \\\"evidence_excerpt\\\": \\\"Chinese paraphrase excerpt: The slope of a line represents the rate of change of the vertical coordinate relative to the horizontal coordinate; given the slope and one point on the line, the point-slope form can be used to construct the equation of the line.\\\"}\", \"structured_conditions\": \"{\\\"question_type\\\": \\\"Algebra: Linear equations and linear models\\\", \\\"given_conditions\\\": [\\\"The line passes through point (2, 5).\\\", \\\"The line passes through point (8, 17).\\\", \\\"Points on the x-axis satisfy y=0.\\\"], \\\"target_quantity\\\": \\\"Find the x-coordinate of the intersection of the line with the x-axis.\\\", \\\"answer_choices\\\": [\\\"-1\u002F2\\\", \\\"1\u002F2\\\", \\\"2\\\", \\\"-2\\\"]}\"}","A line passes through the points (2, 5) and (8, 17). What is the x-coordinate of the point where the line intersects the x-axis?","The problem states that a line passes through (2,5) and (8,17), and asks for its x-intercept. Using the slope formula m=(17−5)\u002F(8−2)=2, the point-slope form gives y−5=2(x−2), i.e., y=2x+1. Points on the x-axis satisfy y=0, so 0=2x+1, x=-1\u002F2. This problem directly applies two-point slope, point-slope form, and the definition of intercept; the applicable preconditions are that the two points have different x-coordinates and do lie on the same line.","0.95","Linear function knowledge material: The slope of a non-vertical line equals the ratio of the difference in y-coordinates to the difference in x-coordinates of two points. A line can usually be written in slope-intercept form y=mx+b, where m is the slope and b is the y-intercept. The intersection with the x-axis satisfies y=0.","{\"relation_type\": \"Direct application\", \"linked_concepts\": [\"Two-point slope\", \"Point-slope form\", \"Slope-intercept form\", \"x-intercept\"], \"reasoning_link\": \"The two points given in the problem directly provide the coordinates needed to calculate the slope. Substituting (2,5) and (8,17) into the slope formula gives m=2, then the point-slope form is used to construct the line equation, and finally the condition y=0 on the x-axis is used to find the intercept.\", \"answer_derivation\": \"m=(17−5)\u002F(8−2)=2. Using point-slope form: y−5=2(x−2), which simplifies to y=2x+1. Setting y=0 gives 0=2x+1, so x=-1\u002F2, corresponding to choice -1\u002F2.\", \"limitation_notes\": \"Do not mistake the x-intercept for the y-intercept; the x-intercept requires y=0, while the y-intercept requires x=0. Also note that the order of coordinate differences in the slope calculation must be consistent.\"}","{\"concept_name\": \"Slope, slope-intercept form, and x-intercept\", \"concept_definition\": \"Given two points with different x-coordinates, the slope of the line is m=(y₂−y₁)\u002F(x₂−x₁). The slope-intercept form of the line is y=mx+b; setting y=0 gives the x-coordinate of its intersection with the x-axis.\", \"formula_or_theorem\": \"m=(y₂−y₁)\u002F(x₂−x₁); y−y₁=m(x−x₁); the x-intercept is found by setting y=0.\", \"applicability_conditions\": [\"The x-coordinates of the two points must be different; otherwise the denominator in the slope formula is zero.\", \"The two points must determine the same line.\", \"When finding the x-intercept, y must be set to 0.\"], \"source_basis\": \"OpenStax, Algebra and Trigonometry 2e, content on slope and equations of lines in the Linear Functions chapter; the source notes and evidence excerpts in this sample are paraphrased in Chinese, not verbatim quotations.\", \"evidence_excerpt\": \"Chinese paraphrase excerpt: The slope of a line represents the rate of change of the vertical coordinate relative to the horizontal coordinate; given the slope and one point on the line, the point-slope form can be used to construct the equation of the line.\"}","{\"question_type\": \"Algebra: Linear equations and linear models\", \"given_conditions\": [\"The line passes through point (2, 5).\", \"The line passes through point (8, 17).\", \"Points on the x-axis satisfy y=0.\"], \"target_quantity\": \"Find the x-coordinate of the intersection of the line with the x-axis.\", \"answer_choices\": [\"-1\u002F2\", \"1\u002F2\", \"2\", \"-2\"]}","{\"question_text\":\"In a circle, AB is a diameter and C is a point on the circle. If AC=9 and BC=12, what is the area of the circle?\",\"retrieval_text\":\"In the circle, AB is a diameter, C lies on the circle, AC=9, and BC=12. By the theorem that an inscribed angle subtended by a diameter is a right angle, triangle ABC is a right triangle with AB as its hypotenuse. By the Pythagorean theorem, AB²=9²+12²=225, so the diameter AB=15 and the radius r=15\\\u002F2. The area of the circle is A=πr²=π(15\\\u002F2)²=225π\\\u002F4. This problem is a combined application of the inscribed angle theorem, the Pythagorean theorem, and the circle area formula.\",\"evidence_quality\":\"0.93\",\"source_knowledge\":\"Circle and right triangle knowledge material: An inscribed angle with a diameter of the circle as one side and another point on the circle as its vertex is a right angle. Therefore, when AB is a diameter and C lies on the circle, triangle ABC is right-angled at C. The diameter AB is the hypotenuse of this right triangle and can be found using the Pythagorean theorem; then divide the diameter by 2 to get the radius, and finally use the circle area formula.\",\"solution_relation\":\"{\\\"relation_type\\\": \\\"Combined application\\\", \\\"linked_concepts\\\": [\\\"An inscribed angle subtended by a diameter is a right angle\\\", \\\"Pythagorean theorem\\\", \\\"Radius of a circle\\\", \\\"Area of a circle\\\"], \\\"reasoning_link\\\": \\\"Because AB is a diameter and C lies on the circle, ∠ACB is a right angle, AC and BC are the two legs, and AB is the hypotenuse. First use the Pythagorean theorem to find the diameter, then convert it to the radius and calculate the area of the circle.\\\", \\\"answer_derivation\\\": \\\"AB²=9²+12²=81+144=225, so AB=15. The radius r=15\\\u002F2. The area of the circle is A=π(15\\\u002F2)²=225π\\\u002F4, so the correct answer is 225π\\\u002F4.\\\", \\\"limitation_notes\\\": \\\"Do not directly assume the triangle is right-angled merely from AC=9 and BC=12; the right angle comes from AB being a diameter and C lying on the circle. Another common mistake is treating 15 as the radius, which incorrectly makes the area four times too large.\\\"}\",\"knowledge_evidence\":\"{\\\"concept_name\\\": \\\"Inscribed angle theorem, Pythagorean theorem, and area of a circle\\\", \\\"concept_definition\\\": \\\"If all three vertices of a triangle lie on the same circle and one side is a diameter of the circle, then the angle subtended by that diameter is a right angle. Therefore, the diameter is the hypotenuse of that right triangle. The side lengths of a right triangle satisfy the Pythagorean theorem, and the area of a circle equals π times the square of the radius.\\\", \\\"formula_or_theorem\\\": \\\"∠ACB=90°; AB²=AC²+BC²; A=πr², and r=AB\\\u002F2.\\\", \\\"applicability_conditions\\\": [\\\"AB must actually be a diameter of the circle, and A, B, and C must all lie on the same circle.\\\", \\\"The Pythagorean theorem can only be used for right triangles.\\\", \\\"The radius of a circle equals half the diameter; the diameter cannot be substituted directly into the circle area formula.\\\"], \\\"source_basis\\\": \\\"CK-12 Geometry FlexBook 2.0, chapters related to Right Triangles and Circles; the source notes and evidence excerpts in this sample are paraphrased in Chinese, not verbatim quotations.\\\", \\\"evidence_excerpt\\\": \\\"Chinese paraphrase excerpt: An inscribed angle subtended by a diameter is a right angle; in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs; the area of a circle is given by πr².\\\"}\",\"structured_conditions\":\"{\\\"question_type\\\": \\\"Geometry: Circles, right triangles, and the Pythagorean theorem\\\", \\\"given_conditions\\\": [\\\"AB is a diameter of the circle.\\\", \\\"Point C lies on the circle.\\\", \\\"AC=9.\\\", \\\"BC=12.\\\", \\\"The formula for the area of a circle is A=πr².\\\"], \\\"target_quantity\\\": \\\"Find the area of the circle.\\\", \\\"answer_choices\\\": [\\\"225π\\\u002F16\\\", \\\"225π\\\u002F4\\\", \\\"225π\\\u002F2\\\", \\\"450π\\\"]}\"}","MOBIUSI INC","https:\u002F\u002Fstatic.mobiusi.com\u002Fmobiwusi\u002F20250829\u002F7bf4ac62b4533e2b809998a6f1c5491e.png?Expires=1792463130&OSSAccessKeyId=LTAI5tH8MQqhgeiAtrzCHrxM&Signature=v6Tr8jBrdm4s5wU1SnnyGsYg%2Bhw%3D&x-oss-process=image%2Fresize%2Cm_fill%2Cw_35%2Ch_35",[687,693],{"type":688,"title":689,"count_data":690},"bar","Length distribution",[691],{"name":692,"value":483},"1k lower",{"type":694,"title":695,"count_data":696},"pie","Language distribution",[697],{"name":698,"value":483},"English"]