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Explain why from E' to G' is 6, and why from F to G is 22.5

The two triangles \( \triangle EFG \) and \( \triangle E'F'G' \) are similar, meaning their corresponding sides are proportional. To solve for the missing values, let's use the side lengths we know and apply the property of similar triangles.

### Step 1: Set up the ratio for corresponding sides
- In the larger triangle \( \triangle EFG \), we know:
  - \( EF = 10 \)
  - \( FG = 15 \)
  
- In the smaller triangle \( \triangle E'F'G' \), we know:
  - \( F'G' = 9 \)
  - \( E'F' = 4 \)

To find the missing sides, we use the fact that the ratios of corresponding sides are the same. Let's find the ratio between \( \triangle EFG \) and \( \triangle E'F'G' \).

\[
\text{Ratio} = \frac{E'F'}{EF} = \frac{4}{10} = 0.4
\]

### Step 2: Find \( E'G' \)
Since \( E'G' \) corresponds to \( EG \), and we know that \( EG = 15 \), we can use the ratio to find \( E'G' \).

\[
E'G' = EG \times 0.4 = 15 \times 0.4 = 6
\]

Thus, the length of \( E'G' \) is **6**.

### Step 3: Find \( FG \)
Now we will use the ratio to find the missing value for \( FG \), which is proportional to \( F'G' = 9 \).

\[
FG = F'G' \div 0.4 = 9 \div 0.4 = 22.5
\]

Thus, the length of \( FG \) is **22.5**.

### Explanation Summary:
- The side \( E'G' \) is **6** because the ratio between \( EF \) and \( E'F' \) is 0.4, and applying this ratio to \( EG \), we get \( 6 \).
- The side \( FG \) is **22.5** because using the same ratio of 0.4, we calculate \( FG \) as \( 9 \div 0.4 \), giving us 22.5.

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:
1. How do you know which sides are corresponding in similar triangles?
2. What would happen if the ratio was different?
3. Can you use another method to verify that \( \triangle EFG \) and \( \triangle E'F'G' \) are indeed similar?
4. What would the perimeter of each triangle be after finding the missing sides?
5. How can the area of each triangle be compared using the ratio of similarity?

**Tip:** When dealing with similar triangles, always check that the ratio of all corresponding sides remains consistent.

Learn how to determine missing side lengths in similar triangles EFG and E'F'G'. This solution uses the property of similar triangles and proportional side lengths to calculate E'G' as 6 and FG as 22.5. Suitable for students in Grades 7-10.

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