Triangles are fun
Geometry
Level
3
In the figure, the areas of Triangle(CEF), Triangle(ABE), and Triangle(ADF) are 3, 4, and 5 respectively. Find the area of Triangle(AEF).
The answer is 8.
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1 solution
By Letting y = AD and x = AB,
Then,
BE = 8/x , CE = y-8/x , CF = 6/(y-8/x) , and DF = 10/y
Therefore,
DF + CF = AB
10/y + 6x/(xy-8) = x
10(xy-8) + 6xy = xy(xy-8)
10xy - 80 + 6xy = (xy)^2 - 8xy
24xy - (xy)^2 = 80
xy(24-xy) = 80
in this case, we need to find the max # that is divisible by 80 where in, xy ≤ 24.
In that case, the value of xy that is only compatible is 20.
So, the area of Rectangle(ABCD) = 20.
So,
Area of Triangle(AEF) = Area of Rectangle(ABCD) - [Area of Triangle(CEF) + Triangle(ABE)+Triangle(ADF)] = 20 - [3+4+5] = 8
Final answer: 8.