Problem Analysis

1. Question 1:

   • We have two similar triangles.
   • Using the property of similarity, corresponding sides are proportional.

   The proportions can be written as: $\frac{12}{6} = \frac{5}{x}$ Solve for $x$.

2. Question 2:

   • Two similar polygons with corresponding side lengths provided.
   • Use the similarity ratio: $\frac{4}{7} = \frac{x}{21}$ Solve for $x$.

Solutions:

1) Solving for $x$ in Question 1:

From the proportion: $\frac{12}{6} = \frac{5}{x} \implies 2 = \frac{5}{x}$ Multiply both sides by $x$: $2x = 5 \implies x = \frac{5}{2} = 2.5 \, \text{cm}.$

2) Solving for $x$ in Question 2:

From the proportion: $\frac{4}{7} = \frac{x}{21}$ Cross multiply: $4 \times 21 = 7 \times x \implies 84 = 7x$ Solve for $x$: $x = \frac{84}{7} = 12 \, \text{cm}.$

Final Answers:

1. $x = 2.5 \, \text{cm}$
2. $x = 12 \, \text{cm}$

Would you like a detailed step-by-step breakdown or have other questions?

Related Questions:

1. What are the properties of similar triangles?
2. How can we generalize proportionality in polygons?
3. How to use ratios effectively for scaling geometric shapes?
4. Can the same method apply for three-dimensional figures?
5. What are other real-life applications of similarity in geometry?

Tip: Always ensure the triangles or polygons are verified as similar before applying proportionality.