Example 2 As shown in Figure 2, given the side lengths of $\triangle D E F$ are $1, \sqrt{3}, 2$, the regular hexagonal grid is composed of 24 equilateral triangles with side length 2. Using the vertices of these equilateral triangles to draw $\triangle A B C$, such that $\triangle A B C \backsim \triangle D E F$. If the similarity ratio $\frac{A B}{D E}=k$, then, the number of different values of $k$ is ( ).
(A) 1
(B) 2
(C) 3
(D) 4
(2008, Shida Cup Jiangsu Province Middle School Mathematics Application and Innovation Invitational Competition Final)
Explanation: By drawing, we can create a triangle similar to $\triangle D E F$. Triangles with different similarity ratios are shown in Figure 3 as three types.
Therefore, there are 3 different values of $k$. Hence, the answer is (C).