To solve for $x$, we use the property that similar rectangles have proportional sides. Let's calculate step-by-step:

1. The ratio of the widths of the two rectangles is given by: $\frac{15}{6x}$

2. The ratio of the heights of the two rectangles is given by: $\frac{x}{10}$

Since the rectangles are similar, these ratios are equal: $\frac{15}{6x} = \frac{x}{10}$

3. Cross-multiply to solve for $x$: $15 \cdot 10 = 6x \cdot x$ $150 = 6x^2$

4. Simplify and solve for $x^2$: $x^2 = \frac{150}{6} = 25$

5. Solve for $x$: $x = \sqrt{25} = 5$

Final Answer:

$x = 5 \, \text{mm}$

Would you like further explanation or help with another question?

Related Questions:

1. How do we calculate the scale factor between two similar figures?
2. What is the geometric significance of similar shapes?
3. Can you solve for $x$ if the rectangles were not similar?
4. What are other real-life applications of proportions in geometry?
5. How can you verify that two shapes are indeed similar?

Tip:

Always check that your ratios are consistent when working with similar figures—this ensures accuracy!