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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 CE = b/4
- Let T be a point on BA and AT = c/4
- Let F be a point of BC and BF = a/4
- Join ET, TF and FE so that make four triangles
- 1. Prove that area of triangle EFT = 7*(area of triangle ABC)/16
- 2. Prove that area FEC = area EAT = area TBF
- 3. Area of triangle EFT = 7*(R^2)*sin(A)*sin(B)*sin(C)/8
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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*(R^2)*sin(A)*sin(B)*sin(C)/8
Proof
- Area FEC
- Area = CE*CF*sin(C)/2
- Area = (b/4)*(3*a/4)*sin(C)/2
- Area = 3*a*b*sin(C)/32
- Area = (12*R^2)*sin(A)*sin(B)*sin(C)/32
- Area = 3*(R^2)*sin(A)*sin(B)*sin(C)/8
- Area EAT
- Area = AE*AT*sin(A)
- Area = (b/4)*(3*c/4)*sin(A)/2
- Area = 3*b*c*sin(A)/32
- Area = (12*R^2)*sin(A)*sin(B)*sin(C)/32
- Area = 3*(R^2)*sin(A)*sin(B)*sin(C)/8
- Area TBF
- Area = BT*BE*sin(B)/2
- Area = (c/4)*(3*a/4)*sin(B)/2
- Area = 3*c*a*sin(B)/32
- Area = (12*R^2)*sin(A)*sin(B)*sin(C)/32
- Area = 3*(R^2)*sin(A)*sin(B)*sin(C)/8
- Area = area ABC - (area FEC + area EAT + area TBF)
- Area = (2 - 3*3/8)*(R^2)*sin(A)*sin(B)*sin(C)
- Area = 7*(R^2)*sin(A)*sin(B)*sin(C)/8
- Area = 7*(area ABC)/16
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Q04. Area EFT = 7*(area ABC)/16
Find area EFT in terms of area ABC
- Area = (1 - 3*(3*(area ABC))/16)
- Area = (1 - 9/16)*(area ABC)
- Area = 7*(area ABC)/16
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