// Numbas version: exam_results_page_options {"name": "MATH6058 Solve for x and y on a given triangle and calculate the area", "extensions": [], "custom_part_types": [], "resources": [["question-resources/q4.png", "/srv/numbas/media/question-resources/q4.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "

i) $\\frac{\\var{lent11}}{\\sin(180^{\\circ}-(\\var{ang11}^{\\circ}+\\var{ang12}^{\\circ}))} = \\frac{x}{\\sin(\\var{ang12}^{\\circ})}$

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$x = \\frac{\\var{lent11}}{\\sin(180^{\\circ}-(\\var{ang11}^{\\circ}+\\var{ang12}^{\\circ}))} \\times \\sin(\\var{ang12}^{\\circ}) = \\var{ans11}mm$

\n

$\\frac{\\var{lent11}}{\\sin(180^{\\circ}-(\\var{ang11}^{\\circ}+\\var{ang12}^{\\circ}))} = \\frac{y}{Sin(\\var{ang11}^{\\circ})}$

\n

$y = \\frac{\\var{lent11}}{Sin(180^{\\circ}-(\\var{ang11}^{\\circ}+\\var{ang12}^{\\circ}))} \\times \\sin(\\var{ang11}^{\\circ}) = \\var{ans12}mm$

\n

$\\frac{1}{2} \\times \\var{ans11} \\times \\var{lent11} \\times \\sin(\\var{ang11}^{\\circ}) = \\var{ans13}mm^2$

\n

\n

ii) $\\frac{\\var{lent21}}{\\sin(\\var{ang21}^{\\circ})} = \\frac{\\var{lent22}}{\\sin(x^{\\circ})}$

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$\\sin(x^{\\circ}) = \\var{lent22} \\div \\frac{\\var{lent21}}{\\sin(\\var{ang21}^{\\circ})}$

\n

$x = \\var{ans21}^{\\circ}$

\n

$y = 180^{\\circ} - (\\var{ans21}^{\\circ}+\\var{ang21}^{\\circ}) = \\var{ans22}^{\\circ}$

\n

$\\frac{\\var{lent21}}{\\sin(\\var{ang21}^{\\circ})} = \\frac{z}{\\sin(\\var{ans22}^{\\circ})}$

\n

$z = \\frac{\\var{lent21}}{\\sin(\\var{ang21})} \\times \\sin(\\var{ans22}) = \\var{ans23}mm$

\n

$\\frac{1}{2} \\times \\var{ans23} \\times \\var{lent22} \\times \\sin(\\var{ang21}^{\\circ}) = \\var{ans24}mm^2$

\n

\n

iii) $\\frac{\\var{lent31}}{\\sin(\\var{ang31}^{\\circ})} = \\frac{x}{\\sin(180^{\\circ}-(\\var{ang31}^{\\circ}+\\var{ang32}^{\\circ}))}$

\n

$x = \\frac{\\var{lent31}}{\\sin(\\var{ang31}^{\\circ})} \\times \\sin(180^{\\circ}-(\\var{ang31}^{\\circ}+\\var{ang32}^{\\circ})) = \\var{ans31}mm$

\n

$\\frac{\\var{lent31}}{\\sin(\\var{ang31}^{\\circ})} = \\frac{y}{\\sin(\\var{ang32}^{\\circ})}$

\n

$y = \\frac{\\var{lent31}}{\\sin(\\var{ang31}^{\\circ})} \\times \\sin(\\var{ang32}^{\\circ}) = \\var{ans32}mm$

\n

$\\frac{1}{2} \\times \\var{ans31} \\times \\var{ans32} \\times \\sin(\\var{ang31}^{\\circ}) = \\var{ans33}mm^2$

\n

\n

iv) $\\frac{\\var{lent31a}}{\\sin(\\var{ang31a}^{\\circ})} = \\frac{x}{\\sin(\\var{ang32a}^{\\circ})}$

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$x = \\frac{\\var{lent31a}}{\\sin(\\var{ang31a}^{\\circ})} \\times \\sin(\\var{ang32a}^{\\circ}) = \\var{ans31a}mm$

\n

$\\frac{\\var{lent31a}}{\\sin(\\var{ang31a}^{\\circ})} = \\frac{y}{\\sin(180^{\\circ}-(\\var{ang32a}^{\\circ}+\\var{ang31a}^{\\circ}))}$

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$y = \\frac{\\var{lent31a}}{\\sin(\\var{ang31a})} \\times \\sin(180-(\\var{ang32a}+\\var{ang31a})) = \\var{ans32a}mm$

\n

$\\frac{1}{2} \\times \\var{ans31a} \\times \\var{ans32a} \\times \\sin(\\var{ang31a}^{\\circ}) = \\var{ans33a}mm^2$

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Solve for x and y on a given triangle and calculate the area

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rebelmaths

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Watch the video below to better understand the sine rule formula

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Video

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Solve the following mensuration questions to 2 decimal places:

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Note: You may need to scroll down to see the diagrams.

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Find the values of x and y:

\n

Note: Keep calculations to 3 decimal places for accuracy but the final answers must be entered to 2 decimal places.

\n

{tri(ans12,h1,lent11,ang11,ang12,w1)}

\n

x = [[0]]mm

\n

y = [[1]]mm

\n

Area = [[2]]$mm^2$

", "type": "gapfill", "steps": [{"marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "prompt": "

First you need to find the third angle. Remember that the three angles of any triangle add up to 180 degrees. Next, pair up opposite sides and angles to use the sine rule.

", "type": "information", "variableReplacements": []}], "variableReplacements": []}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "prompt": "

Find the values of x and y:

\n

Note: Keep calculations to 3 decimal places for accuracy but the final answers must be entered to 2 decimal places.

\n

{tri2(w2,h2,ans23,lent21,lent22,ang21)}

\n

x = [[0]] $^{\\circ}$

\n

y = [[1]] $^{\\circ}$

\n

z = [[2]]mm

\n

Area = [[3]]$mm^2$

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Find the values of x and y:

\n

Note: Keep calculations to 3 decimal places for accuracy but the final answers must be entered to 2 decimal places.

\n

{tri3(lent31,h3,ang31,ang32,l2)}

\n

x = [[0]]mm

\n

y = [[1]]mm

\n

Area = [[2]]$mm^2$

", "type": "gapfill", "gaps": [{"marks": 1, "maxValue": "{ans31}+0.5", "showCorrectAnswer": true, "precision": "2", "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "mustBeReducedPC": 0, "type": "numberentry", "allowFractions": false, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "variableReplacements": [], "showPrecisionHint": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "minValue": "{ans31}-0.5", "precisionType": "dp"}, {"marks": 1, "maxValue": "{ans32}+0.5", "showCorrectAnswer": true, "precision": "2", "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "mustBeReducedPC": 0, "type": "numberentry", "allowFractions": false, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "variableReplacements": [], "showPrecisionHint": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "minValue": "{ans32}-0.5", "precisionType": "dp"}, {"marks": 1, "maxValue": "{ans33}+1", "showCorrectAnswer": true, "precision": "2", "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "mustBeReducedPC": 0, "type": "numberentry", "allowFractions": false, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "variableReplacements": [], "showPrecisionHint": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "minValue": "{ans33}-1", "precisionType": "dp"}], "variableReplacements": []}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "prompt": "

Find the values of x and y:

\n

Note: Keep calculations to 3 decimal places for accuracy but the final answers must be entered to 2 decimal places.

\n

{tri4(lent31a,ans31a,ans32a,ang31a,ang32a,h3a,l2a)}

\n

x = [[0]]mm

\n

y = [[1]]mm

\n

Area = [[2]]$mm^2$

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"definition": "random(55..(ang11-8))", "name": "ang12", "description": "", "templateType": "anything"}, "ang31": {"group": "Ungrouped variables", "definition": "random(75..85)", "name": "ang31", "description": "", "templateType": "anything"}, "ang32a": {"group": "Q4", "definition": "random(47..55)", "name": "ang32a", "description": "", "templateType": "anything"}, "h1": {"group": "Ungrouped variables", "definition": "precround((lent11/(sin(radians(90))))*sin(radians(ang12)),2)", "name": "h1", "description": "", "templateType": "anything"}, "h3a": {"group": "Q4", "definition": "precround((ans32a/(sin(radians(90))))*sin(radians(ang32a)),2)", "name": "h3a", "description": "", "templateType": "anything"}, "w1": {"group": "Ungrouped variables", "definition": "precround((ans12-(lent11/(sin(radians(90))))*sin(radians(90-ang12))),2)", "name": "w1", "description": "", "templateType": "anything"}, "ang31a": {"group": "Q4", "definition": "random(71..85)", "name": "ang31a", "description": "", "templateType": "anything"}, "ans32": {"group": "Ungrouped variables", "definition": "precround((lent31/(sin(radians(ang31))))*sin(radians(ang32)),2)", "name": "ans32", "description": "", "templateType": "anything"}, "h3": {"group": "Ungrouped variables", "definition": "precround((ans32/(sin(radians(90))))*sin(radians(180-(ang31+ang32))),2)", "name": "h3", "description": "", "templateType": "anything"}, "ans12": {"group": "Ungrouped variables", "definition": "precround((lent11/(sin(radians(180-(ang11+ang12)))))*sin(radians(ang11)),2)", "name": "ans12", "description": "", "templateType": "anything"}, "w2": {"group": "Ungrouped variables", "definition": "precround((lent21/(sin(radians(90))))*sin(radians(90-ans21)),2)", "name": "w2", "description": "", "templateType": "anything"}, "lent31a": {"group": "Q4", "definition": "random(72..98#2)", "name": "lent31a", "description": "", "templateType": "anything"}, "ans11": {"group": "Ungrouped variables", "definition": "precround((lent11/(sin(radians(180-(ang11+ang12)))))*sin(radians(ang12)),2)", "name": "ans11", "description": "", "templateType": "anything"}, "l2a": {"group": "Q4", "definition": "precround((ans32a/(sin(radians(90))))*sin(radians(180-(ang32a+90))),2)", "name": "l2a", "description": "", "templateType": "anything"}, "lent11": {"group": "Ungrouped variables", "definition": "random(35..45)", "name": "lent11", "description": "", "templateType": "anything"}, "ans24a": {"group": "Ungrouped variables", "definition": "precround((0.5*(w2)*h2),2)", "name": "ans24a", "description": "", "templateType": "anything"}, "l2": {"group": "Ungrouped variables", "definition": "precround((ans31/(sin(radians(90))))*sin(radians(90-ang32)),2)", "name": "l2", "description": "", "templateType": "anything"}, "ans33a": {"group": "Q4", "definition": "precround((0.5*lent31a*h3a),2)", "name": "ans33a", "description": "", "templateType": "anything"}, "ang11": {"group": "Ungrouped variables", "definition": "random(63..76)", "name": "ang11", "description": "", "templateType": "anything"}, "lent31": {"group": "Ungrouped variables", "definition": "random(101..120)", "name": "lent31", "description": "", "templateType": "anything"}, "lent22": {"group": "Ungrouped variables", "definition": "random(35..50)", "name": "lent22", "description": "", "templateType": "anything"}, "ans13": {"group": "Ungrouped variables", "definition": "precround((0.5*ans12*h1),2)", "name": "ans13", "description": "", "templateType": "anything"}}, "type": "question", "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}]}], "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}