Question

triangle efg with sides ef=10, eg=15, fg=27.5 (and another figure with e and side 4, but focus on the triangle efg with labeled sides 10, 15, 27.5 and vertices e, f, g).

Explanation:

Assuming we need to find the scale factor or related length (maybe for similar triangles, as there's another triangle with side 4). Let's check the sides. The original triangle has sides 10, 15, 22.5? Wait, maybe the other triangle (with \( E' \), \( F' \), \( G' \)) has a side 4 corresponding to 10 or 15. Wait, let's assume similarity. Let's take side \( EF = 10 \) and corresponding side \( E'F' = 4 \). Then scale factor \( k=\frac{4}{10}=\frac{2}{5} \). If we need to find a corresponding side, say for \( EG = 15 \), the corresponding side would be \( 15\times\frac{2}{5}=6 \), or for \( FG = 22.5 \), corresponding side \( 22.5\times\frac{2}{5}=9 \). But maybe the problem is to find the scale factor or a missing side. Let's proceed with similarity.

Step1: Identify Corresponding Sides

Assume triangle \( EFG \) ~ triangle \( E'F'G' \), with \( EF = 10 \) and \( E'F' = 4 \).

Step2: Calculate Scale Factor

Scale factor \( k=\frac{E'F'}{EF}=\frac{4}{10}=\frac{2}{5} \).

Step3: Find Corresponding Side (e.g., for \( EG = 15 \))

Corresponding side \( E'G' = EG\times k = 15\times\frac{2}{5}=6 \). Or for \( FG = 22.5 \), \( F'G' = 22.5\times\frac{2}{5}=9 \).

(Note: Since the problem isn't fully stated, but based on the diagram, likely similarity. If the question is about scale factor, it's \( \frac{2}{5} \), or if finding a missing side, use the scale factor.)

Answer:

If scale factor (from \( EFG \) to \( E'F'G' \)): \( \frac{2}{5} \) (or 0.4). If finding \( E'G' \) (corresponding to \( EG = 15 \)): 6. If finding \( F'G' \) (corresponding to \( FG = 22.5 \)): 9. (Depends on the exact question, but assuming similarity with \( EF = 10 \) and \( E'F' = 4 \), the scale factor is \( \frac{2}{5} \).)