The two shapes in the image are pentagons, and they appear to be similar polygons, which means their corresponding sides are proportional.

Given:

• The side lengths of the smaller pentagon are $6 \, \text{cm}$, $9 \, \text{cm}$, and the corresponding sides in the larger pentagon are $9 \, \text{cm}$ and $x \, \text{cm}$.

Step-by-step solution:

1. First, identify corresponding sides. The $6 \, \text{cm}$ side of the smaller pentagon corresponds to the $9 \, \text{cm}$ side of the larger pentagon.

2. Set up a proportion using the corresponding sides: $\frac{9}{6} = \frac{x}{9}$

3. Solve for $x$: $\frac{9}{6} = \frac{x}{9}$ Cross-multiply: $9 \times 9 = 6 \times x$ $81 = 6x$ Divide both sides by 6: $x = \frac{81}{6} = 13.5$

Thus, the value of $x$ is 13.5 cm, so the correct answer is B.

Would you like further details or explanations on any step?

Related questions:

1. What is the perimeter of each pentagon?
2. How do you find the area of a regular pentagon?
3. How can you determine whether two polygons are similar?
4. If a pentagon has side lengths increasing by a certain ratio, how do you find the ratio?
5. How do you calculate the angles in a regular pentagon?

Tip:

Always check if two polygons are similar before applying proportions; similarity requires both equal angles and proportional corresponding sides.