The area of similar triangles is proportional to the square of their corresponding linear dimensions. Given that the linear scale factor between the triangles is \( \frac{2}{3} \), we can calculate the ratio of their areas.
Let \( A_1 \) be the area of the original triangle and \( A_2 \) be the area of the similar triangle. The relationship between the areas can be expressed as:
\[ \frac{A_1}{A_2} = \left(\frac{\text{side of triangle 1}}{\text{side of triangle 2}}\right)^2 = \left(\frac{2}{3}\right)^2 \]
Calculating \( \left(\frac{2}{3}\right)^2 \):
\[ \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]
This means the ratio of the areas is:
\[ \frac{12}{A_2} = \frac{4}{9} \]
To find \( A_2 \), we can set up the equation:
\[ 12 \cdot 9 = 4 \cdot A_2 \]
\[ 108 = 4 A_2 \]
Now, divide both sides by 4:
\[ A_2 = \frac{108}{4} = 27 \]
Thus, the area of the triangle similar to the one with an area of 12 square cm is
\[ \boxed{27} \text{ square cm.} \]
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