[Math] Find area of shaded regions in a triangle

areageometryratiotriangles

I am trying to solve this problem.

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In $\triangle ABC$, $CD=3BD$ and $DE=AE$. Given that the area of $\triangle ABC$ is 14$cm^2$. Find the total area, in $cm^2$, of the shaded regions.

I divided the triangle into two parts and found the values of the area for both triangles.

How should I proceed and use the given to solve the problem?

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Best Answer

By applying a linear trasformation to the triangle you won't change the proprtion between the shaded area and the area of the entire triangle, so you can reduce the problem to computing the areas in a soecial case (one that makes the computation easy) and then apply the proportion to get the final result.

For example, suppose $|AB|=|BC|=4$ and that there is a right angle on $B$. Then you get that the shaded area is $24/7$, while the total area is $8$. You can then conclude that the answer to your problem is $\frac{3}{7}14=6$.

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