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We are given that triangles K and L are similar. This means that the corresponding sides of the triangles are proportional.

The side lengths of triangle K are:
- 6 mm
- 10 mm
- 11 mm

The side lengths of triangle L are:
- 30 mm
- 55 mm
- \( x \) mm (which we need to find).

To set up the proportion, we can match the corresponding sides of the triangles. The side that corresponds to the unknown side \( x \) is the side of length 10 mm in triangle K, and the side corresponding to 55 mm is the side of 11 mm in triangle K.

Using the proportion of the corresponding sides:

\[
\frac{x}{10} = \frac{55}{11}
\]

Now, solving for \( x \):

\[
\frac{x}{10} = 5 \quad \text{(since \( \frac{55}{11} = 5 \))},
\]

\[
x = 10 \times 5 = 50 \text{ mm}.
\]

So, the value of \( x \) is 50 mm.

Would you like further details or have any questions? 

Here are some related questions:
1. How do you determine corresponding sides in similar triangles?
2. What are the properties of similar triangles?
3. How do you simplify ratios when solving proportions?
4. What is the significance of proportionality in geometry?
5. How do you verify if two triangles are similar?

**Tip:** Always ensure that corresponding sides are matched correctly when working with similar triangles. This helps maintain the correct proportion throughout your calculations.

Triangles K and L are similar. What is the value of x? Give your answer as an integer or as a fraction in its simplest form. Triangle K has sides 6 mm, 10 mm, and 11 mm. Triangle L has sides 30 mm, 55 mm, and x mm.

Determine the value of x in the similar triangles K and L using proportions. This problem involves calculating the unknown side length in similar triangles using the ratio of corresponding sides. Ideal for students in Grades 8-10.

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