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Let AC be Perpendicular to CE. Connect A to the midpoint D of CE and connect E to the midpoint B of AC. AD and EB intersect at point F, and BC = 7 inches, CD=9 inches. Find the area of DFE in square inches.

 
 Feb 23, 2021
 #1
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Firstly, you can always put it on a grid - use coordinates.

 

My Solution is synthetic, and uses the mass points method:

Suppose $C$ has mass $1$. We get $A$ to have mass $1$ as well. Therefore, $B$ has mass $2$, and $D$ has mass $2$. Since $A$ has mass $1$ and $D$ has mass $2$, $FD=\frac{1}{2}AF\implies FD=\frac{1}{3}AD$, or the height of $\triangle FDE$ is $\frac{1}{3}$ of the height of the big triangle. The base is $\frac{1}{2}$ of the big triangle. Therefore, the area is $\frac{1}{2}*\frac{1}{3}=\frac{1}{6}$ of the big triangle. The big triangle has area $\frac{1}{2}\cdot 14\cdot 18=126$, so the answer is $\boxed{21}$.

 
 Feb 23, 2021

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