Similar Quadrilaterals: Finding Original Area When Scaled Area is 12

To solve this problem, we'll follow these steps:

• Step 1: Verify the similarity condition and note given areas and similarity ratio.

• Step 2: Calculate the similarity ratio of the sides between the two polygons.

• Step 3: Determine the area ratio using the square of the linear ratio.

• Step 4: Use the area ratio to find the area of polygon $ABCD$.

Now, let's work through each step:
Step 1: We know $S_{A'B'C'D'} = 12$ and the similarity ratio as $\frac{4}{10} = 0.4$.
Step 2: The ratio of corresponding sides is $\frac{A'B'}{AB} = \frac{4}{10} = 0.4$.
Step 3: The area ratio is the square of the side ratio: $(0.4)^2 = 0.16$.
Step 4: Let $S_{ABCD}$ be the area of polygon $ABCD$. Then we have $\frac{12}{S_{ABCD}} = 0.16$. Solving for $S_{ABCD}$, we get:

$\begin{aligned} S_{ABCD} &= \frac{12}{0.16} \\ &= 12 \times \frac{100}{16} \\ &= 75. \end{aligned}$

Therefore, the solution to the problem is $S_{ABCD} = 75$.