In this image, we are given three similar triangles, $\triangle ABC$, $\triangle EFD$, and $\triangle GHI$, and we are tasked with finding unknown side lengths using the property of similar triangles. The sides of similar triangles are proportional, meaning that the ratios of corresponding side lengths in the triangles will be the same.

Let's define the key points:

1. Since all the triangles are similar, the ratio of corresponding sides of these triangles will be equal.
2. We are given some side lengths and need to find the missing side lengths using these proportions.

From the image:

• For $\triangle ABC$, we are given two side lengths (8 and 12). We will need to find the third side.
• Similarly, for $\triangle EFD$, the side lengths given are 5 and an unknown length.
• For $\triangle GHI$, we know the side lengths 6 and 9.

Let's proceed with setting up the ratios:

We can set up the proportions between the corresponding sides of $\triangle ABC$, $\triangle EFD$, and $\triangle GHI$. If we let the unknown sides be represented by variables (for example, $x$ and $y$ for the unknowns), we can solve these proportions accordingly.

I'll now compute this for you.The missing side length $x$ for triangle $\triangle EFD$ is 12.

Would you like further explanation or details about this solution? Here are some related questions that may help:

1. How do we determine which sides are corresponding in similar triangles?
2. Why do the ratios of corresponding sides remain the same in similar triangles?
3. Can we use this technique to solve for angles in similar triangles as well?
4. What are some real-world applications of similar triangles?
5. How do we find the area of a triangle if we know the side lengths?

Tip: When working with proportions, always make sure to match corresponding sides and be consistent in setting up the ratios.