// Numbas version: finer_feedback_settings
{"name": "Pythagorean Relationships", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Screenshot 2025-08-15 141835.png", "Screenshot_2025-08-15_141835.png"], ["question-resources/Screenshot 2025-08-15 141855.png", "Screenshot_2025-08-15_141855.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Pythagorean Relationships", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Pythagoras&rsquo; theorem states that for any right-angled triangle <strong>c</strong><sup>2</sup> = <strong>b</strong><sup>2</sup> + <strong>a</strong><sup>2</sup>. These are labelled on the triangle below. <strong>c</strong> is the longest side of the triangle, or hypotenuse.<br/>We can use this relationship between side lengths to work out missing values, for example in biomechanics we can calculate resultant take-off velocity for a given horizontal and vertical velocity at take-off, if we assume the hypotenuse is our resultant velocity.</p>\n<p></p>\n<p><img src=\"resources/question-resources/Screenshot 2025-08-15 141835.png\"/></p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 20#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1 .. 20#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "c_2": {"name": "c_2", "group": "Ungrouped variables", "definition": "trunc(sqrt({a_2}^2+{b_2}^2),1)", "description": "", "templateType": "anything", "can_override": false}, "a_2": {"name": "a_2", "group": "Ungrouped variables", "definition": "random(1 .. 20#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "b_2": {"name": "b_2", "group": "Ungrouped variables", "definition": "random(1 .. 20#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "a_3": {"name": "a_3", "group": "Ungrouped variables", "definition": "random(1 .. 20#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "c_3": {"name": "c_3", "group": "Ungrouped variables", "definition": "trunc(sqrt({a_3}^2+{b_3}^2),1)", "description": "", "templateType": "anything", "can_override": false}, "b_3": {"name": "b_3", "group": "Ungrouped variables", "definition": "random(1 .. 20#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "a_4": {"name": "a_4", "group": "Ungrouped variables", "definition": "random(1 .. 10#0.1)", "description": "", "templateType": "randrange", "can_override": false}, "b_4": {"name": "b_4", "group": "Ungrouped variables", "definition": "random(1 .. 5#0.1)", "description": "", "templateType": "randrange", "can_override": false}, "c_5": {"name": "c_5", "group": "Ungrouped variables", "definition": "trunc(sqrt({a_5}^2+{b_5}^2),1)", "description": "", "templateType": "anything", "can_override": false}, "b_5": {"name": "b_5", "group": "Ungrouped variables", "definition": "random(1 .. 6#0.1)", "description": "", "templateType": "randrange", "can_override": false}, "a_5": {"name": "a_5", "group": "Ungrouped variables", "definition": "random(1 .. 10#0.1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "a_2", "b_2", "c_2", "a_3", "c_3", "b_3", "a_4", "b_4", "c_5", "b_5", "a_5"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>If b={b}m, and a = {a}m, what is the length of the hypotenuse?</p>", "minValue": "sqrt({a}^2+{b}^2)", "maxValue": "sqrt({a}^2+{b}^2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>If a={a_2}m, and c = {c_2}m, what is the length of <strong>b</strong>?</p>", "minValue": "sqrt({c_2}^2-{a_2}^2)", "maxValue": "sqrt({c_2}^2-{a_2}^2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>If b={b_3}m, and c = {c_3}m, what is the length of <strong>a</strong>?</p>", "minValue": "sqrt({c_3}^2-{b_3}^2)", "maxValue": "sqrt({c_3}^2-{b_3}^2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Now consider that the sides of the triangle each represent a velocity. Side &lsquo;<strong>a</strong>&rsquo; represents the amount of vertical velocity an object has, side &lsquo;<strong>b</strong>&rsquo; is the horizontal velocity, and the hypotenuse, side &lsquo;<strong>c</strong>&rsquo;, is the resultant velocity.<br/>Let&rsquo;s say we kick a football. We impart both horizontal and vertical velocity components to it, which contribute to resultant velocity. Imagine that the horizontal velocity we impart is 3 m/s, and the vertical is 2 m/s. What is the resultant velocity (<strong>V</strong>) of the ball?</p>\n<p>First we draw it out as a triangle:</p>\n<p><img src=\"resources/question-resources/Screenshot 2025-08-15 141835.png\"/>&nbsp;</p>\n<p>Becomes</p>\n<p><img src=\"resources/question-resources/Screenshot 2025-08-15 141855.png\"/></p>\n<p>Use Pythagoras to write out the problem and calculate &lsquo;<strong>V</strong>&rsquo;:<br/><strong>V</strong><sup>2</sup> = 22 + 32<br/><strong>V</strong><sup>2</sup> = 4 + 9 = 13<br/><strong>V</strong>= &radic;13<br/><strong>V</strong>=3.6 m/s</p>\n<p></p>"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A footballer runs at {a_4} m/s and then jumps to avoid a tackle. The vertical velocity of the jump was {b_4} m/s, what was the resultant velocity at the time of the jump?</p>", "minValue": "sqrt({a_4}^2+{b_4}^2)", "maxValue": "sqrt({a_4}^2+{b_4}^2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A long-jumper takes off with a resultant velocity of {c_5} m/s. Their vertical velocity at take-off was {b_5} m/s. What running velocity was required at take-off to achieve the resultant take-off velocity?</p>", "minValue": "sqrt({c_5}^2-{b_5}^2)", "maxValue": "sqrt({c_5}^2-{b_5}^2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Matthew Ellison", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/31197/"}], "resources": ["question-resources/Screenshot 2025-08-15 141835.png", "question-resources/Screenshot 2025-08-15 141855.png"]}]}], "contributors": [{"name": "Matthew Ellison", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/31197/"}]}