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Symbol Defintion
Example : Sqr(x) = square root of x
Q01. Diagram and question
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- Draw a triangle ABC and sides are a,b,c
- Let E be a point on AC and AE = b/3
- Let T be a point on BA and BT = c/4
- Let F be a point on BC and CF = a/2
- Join ET, TF and FE so that make four triangles
- Prove that area of triangle EFT = 7*(area of triangle ABC)/24
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Q02. Area of triangle and sine law
Sine law
- a = 2*R*sin(A)
- b = 2*R*sin(B)
- c = 2*R*sin(C)
- = b*c*sin(A)/2
- = c*a*sin(B)/2
- = a*b*sin(B)/2
- = b*c*sin(A)/2 = b*c*(a/(2*R))/2 = a*b*c/(4*R)
- = c*a*sin(B)/2 = c*a*(b/(2*R))/2 = a*b*c/(4*R)
- = a*b*sin(B)/2 = a*b*(c/(2*R))/2 = a*b*c/(4*R)
- Hence area ABC = 2*(R^2)*sin(A)*sin(B)*sin(C)
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Q03. Area EFT = 7*(area ABC)/24
Construction
- Draw a triangle ABC and sides are a,b,c
- Let E be a point on AC and EA = b/3
- Let T be a point on BA and BT = c/4
- Let F be a point on BC and FB = a/2
- Join ET, TF and FE so that make four triangles
- Find the area of triangle ETF = ?
- Area of triangle CEF
- Area = (FC*CE)*sin(C)/2
- Area = (a/2)*(2*b/3)*sin(C)/2
- Area = (1/3)*a*b*sin(C)/2
- Area = (Area ABC)/3
- Area = (AE*AT)*sin(A)/2
- Area = (b/3)*(3*c/4)*sin(A)/2
- Area = (1/4)*b*c*sin(A)/2
- Area = (Area ABC)/4
- Area = (BT*BF)*sin(B)/2
- Area = (c/4)*(a/2)*sin(B)/2
- Area = (1/8)*b*c*sin(B)/2
- Area = (Area ABC)/8
- Area = ABC - CEF - AET - BFT
- Area = (area ABC)*(1 - 1/3 - 1/4 - 1/8)
- Area = (area ABC)*(7/24)
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