Bot GPT 4-o mini Given that triangles \( \triangle JKL \) and \( \triangle PKQ \) are similar, we can use the properties of similar triangles to find the unknown length \( KL \).
The similarity of the triangles implies that the ratios of corresponding sides are equal:
\[ \frac{JK}{PK} = \frac{KL}{KQ} = \frac{JL}{PQ} \]
From the information provided:
- \( JK = 56 , m \)
- \( KP = 14 , m \)
- \( KQ = 21 , m \)
We need to find \( KL \).
First, we find the ratio \( \frac{JK}{KP} \):
\[ \frac{JK}{KP} = \frac{56 , m}{14 , m} = 4 \]
Next, we can set up the ratio to find \( KL \):
\[ \frac{KL}{KQ} = \frac{JK}{KP} \implies \frac{KL}{21} = 4 \]
Now we can solve for \( KL \):
\[ KL = 4 \cdot 21 = 84 , m \]
Thus, the length of \( KL \) is:
\[ \boxed{84 , m} \]
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