ds","topicId":"80a3a6c4-d669-40f2-94f3-3da2502a5ce1","topicSlug":"area-of-a-triangle","cards":[{"id":"6a7162fb-2470-4366-ad9d-a1b6cb1c98d8","cardType":"basic","front":"Which angle must be used with two chosen sides in the sine area formula?","back":"The included angle between the two sides.","points":1,"mcqOptions":null},{"id":"83016144-d71c-4e23-91d6-e043f1bc5390","cardType":"cloze","front":"For any triangle, $\\text{[...]}$.","back":"For any triangle, $\\boxed{\\text{Area}=\\frac12\\times\\text{base}\\times\\text{perpendicular height}}$.","points":null,"mcqOptions":null},{"id":"adf9b2a8-d6da-4ed9-a478-be67941c72d5","cardType":"basic","front":"What condition must a triangle's height satisfy relative to its chosen base?","back":"It must meet the base at a right angle.","points":1,"mcqOptions":null},{"id":"02fb0232-c2f7-414e-970e-f1571470eae1","cardType":"basic","front":"A side $b$ makes angle $C$ with the base. What is the perpendicular height?","back":"$h=b\\sin C$","points":1,"mcqOptions":null},{"id":"2879fbbd-61a8-454d-8635-e635ff3c1806","cardType":"basic","front":"Why does $b\\sin C$ give the triangle's height?","back":"It is the perpendicular component of side $b$.","points":2,"mcqOptions":null},{"id":"f3487b62-db70-47de-a964-186a5057d0dc","cardType":"cloze","front":"If sides $a$ and $b$ enclose angle $C$, then $\\text{[...]}$.","back":"If sides $a$ and $b$ enclose angle $C$, then $\\boxed{\\text{Area}=\\frac12ab\\sin C}$.","points":null,"mcqOptions":null},{"id":"c14a27e9-d346-42be-af1e-37122e31f88f","cardType":"basic","front":"Which area formula uses sides $b$ and $c$?","back":"$\\text{Area}=\\frac12bc\\sin A$","points":1,"mcqOptions":null},{"id":"df65dc6f-7b3c-4808-8dd3-a67d958b0b60","cardType":"basic","front":"Which area formula uses sides $c$ and $a$?","back":"$\\text{Area}=\\frac12ca\\sin B$","points":1,"mcqOptions":null},{"id":"8054a194-25af-4a22-93a6-be21115d5bc6","cardType":"basic","front":"Two sides are 7 cm and 12 cm with included angle 53°. Find the area.","back":"$33.5\\text{ cm}^2$","points":2,"mcqOptions":null},{"id":"fc2cc51b-2a5b-47b7-a41d-dd1282b11540","cardType":"basic","front":"What units should be used for a triangle's area?","back":"Square units, such as $\\text{cm}^2$ or $\\text{m}^2$.","points":1,"mcqOptions":null},{"id":"5bc89b50-521f-4e7f-9d1e-5587bedbf53b","cardType":"basic","front":"Which calculator mode is required when angles include a degree symbol?","back":"Degree mode","points":1,"mcqOptions":null},{"id":"2295a2f3-fd7b-4460-a717-8294126e5a55","cardType":"basic","front":"Does $\\frac12ab\\sin C$ work when $C$ is obtuse?","back":"Yes - the perpendicular may meet the extended base.","points":2,"mcqOptions":null},{"id":"2f96a3e7-d02a-44b1-994b-27f3e7e0c443","cardType":"basic","front":"Why do 125° and 55° produce the same area for fixed side lengths?","back":"They have the same sine value: $\\sin125^\\circ=\\sin55^\\circ$.","points":2,"mcqOptions":null},{"id":"67999b8e-9359-4592-a7e0-96a773b59a71","cardType":"basic","front":"Two sides are 8 m and 11 m with included angle 125°. Find the area.","back":"$36.0\\text{ m}^2$","points":2,"mcqOptions":null},{"id":"5ca2b55f-f385-42fa-a3da-84df820a085e","cardType":"cloze","front":"From $A=\\frac12ab\\sin C$, the missing side is $\\text{[...]}$.","back":"From $A=\\frac12ab\\sin C$, the missing side is $\\boxed{b=\\frac{2A}{a\\sin C}}$.","points":null,"mcqOptions":null},{"id":"014115db-5251-4fc9-ac33-5752f140521a","cardType":"basic","front":"A triangle has area 48 cm²; sides $x$ cm and 10 cm enclose 42°. Find $x$.","back":"$x\\approx14.3\\text{ cm}$","points":2,"mcqOptions":null},{"id":"609bc828-3c91-489e-b773-39963cbc8eda","cardType":"cloze","front":"Given area $A$ and sides $a$ and $b$, $\\text{[...]}$.","back":"Given area $A$ and sides $a$ and $b$, $\\boxed{\\sin C=\\frac{2A}{ab}}$.","points":null,"mcqOptions":null},{"id":"a9

1.7.5 Area of a triangle Flashcards

1.7.5 Area of a triangle Flashcards