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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 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
- 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. Find BT : TA if (area BTF)^2 = (area CEF)*(area AET)
Other condition
- Let E be a point on AC and CE = 2*b/3
- Let F be a mid point of CB and BF = a/2
- Draw a triangle ABC and sides are a,b,c
- a = BC
- b = CA
- c = AB
- Let E be a point on AC and CE = 2*b/3
- Let F be a mid point of CB and BF = a/2
- Let T be a point on BA
- Join ET, TF and FE so that triangle make four triangles
- Let area of triangle BTF = x
- Let area of triangle EFC = y
- Let area of triangle ATE = z
- 1. If x^2 = y*z, find the ratio BT : TA
- 2. Find the area of triangle ETF
- Area of triangle BTF = x
- x = BT*BF*sin(B)/2
- Since BF = a/2, hence x = BT*a*sin(B)/4
- Area of triangle EFC = y
- y = FC*CE*sin(B)/2
- Since FC = a/2 and CE = 2*b/3, hence y = a*b*sin(C)/6
- Area of triangle ATE = z
- z = AT*AE*sin(A)/2
- Since AE = b/3, hence z = AT*b*sin(A)/6
- Find y*z
- y*z = (a*b*sin(C)/6)*(AT*b*sin(A)/6)
- y*z = AT*a*(b^2)*sin(A)*sin(C)/36
- y*z = AT*a*b*(2*R*sin(B))*sin(A)*sin(C)/36
- y*z = AT*a*b*R*sin(A)*sin(B)*sin(C)/18
- Find x^2
- x^2 = (BT*a*sin(B)/4)^2
- x^2 = (BT^2)*(a^2)*(sin(B)^2)/16
- Find x^2 = y*z
- (BT^2)*(a^2)*(sin(B)^2)/16 = AT*a*b*R*sin(A)*sin(B)*sin(C)/18
- Let BT = u and AT = v
- (u^2)*a*sin(B)/16 = v*b*R*sin(A)*sin(C)/18
- Since a = 2*R*sin(A), b = 2*R*sin(B) and c = 2*R*sin(C)
- (u^2)*2*R*sin(A)*sin(B)/16 = v*b*R*sin(A)*sin(C)/18
- (u^2)*sin(B)/8 = v*b*sin(C)/18
- (u^2)*sin(B)/8 = v*2*R*sin(B)*sin(C)/18
- (u^2)/8 = v*R*sin(C)/9 = v*c/18
- Hence u^2 = 4*v*c/9
- Solve
- u^2 = 4*v*c/9 ............ (1)
- u + v = c ................ (2)
- Find u and v
- (c - v)^2 = 4*v*c/9
- v^2 - 2*v*c + c^2 = 4*v*c/9
- v^2 - 22*v*c/9 + c^2 = 0
- Hence v = ((22c/9) + Sqr((22/9)2 - 4*1*1)*c)/2
- v/c = (2.4444 + Sqr(1.9753086))/2 or v/c = (2.4444 - Sqr(1.9753086))/2
- v/c = (2.4444 + 1.4054567)/2
- v = 1.924948*c (v should be less than c)
- Hence v/c = (2.44444 - 1.4054567)/2
- v = 0.519492*c
- u = 0.480508*c
- Hence BT : TA = u : v = 0.480508 : 0.519492
- Find y*z = AT*a*b*R*sin(A)*sin(B)*sin(C)/18
- = 0.519492*c*a*b*R*sin(A)*sin(B)*sin(C)/18
- = 0.519492*(8*R^4)*(sin(A)^2)*(sin(B)^2)*(sin(C)^2)/18
- = 0.519492*4*(R^4)*(sin(A)^2)*(sin(B)^2)*(sin(C)^2)/9
- = 0.23089*(R^4)*(sin(A)^2)*(sin(B)^2)*(sin(C)^2)
- Find x^2 = ((c^2)*0.480508^2)*(a^2)*(sin(B)^2)/16
- = (0.480508^2)*(4*R^2)*(sin(C)^2)*(4*R^2)*(sin(A)^2)*(sin(B)B^2)/16
- = (0.480508^2)*(R^4)*(sin(A)^2)*(sin(B)B^2)*(sin(C)^2)
- = 0.23089*(R^4)*(sin(A)^2)*(sin(B)^2)*(sin(C)^2)
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Q04. Find area EFT
Find area of triangle EFT
- z = AT*b*sin(A)/6
- z = (0.519492)*4*(R^2)*sin(A)*sin(B)*sin(C)/6
- z = 0.346328(R^2)*sin(A)*sin(B)*sin(C)
- y = a*b*sin(C)/6
- y = 4*(R^2)*sin(A)*sin(B)*sin(C)/6
- y = 0.666666*(R^2)*sin(A)*sin(B)*sin(C)
- Area of ABC = 2*(R^2)*sin(A)*sin(B)*sin(C)
- Area of EFT = Area ABC - x - y - z
- = (2 - 0.480508 - 0.666666 - 0.346328)*(R^2)*sin(A)*sin(B)*sin(C)
- = (0.506498)*(R^2)*sin(A)*sin(B)*sin(C)
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