Calculate the value of x given two similar triangles with side lengths.

1. Identify similar triangles: The problem states that the triangles are similar. This means their corresponding angles are equal, and the ratio of their corresponding sides is constant.

1. Identify corresponding sides: From the diagram, we can see that the horizontal line segment of length 9 cm is parallel to the base of the larger triangle, which has length 24.3 cm. This parallel line creates a smaller triangle within the larger one. The side of length 6 cm is the top part of the vertical side of the larger triangle, and $x$ is the bottom part. Therefore, the side of length 6 cm corresponds to the side of length $x$, and the side of length 9 cm corresponds to the side of length 24.3 cm.

3. Set up a proportion: Since the triangles are similar, the ratio of corresponding sides is equal. We can set up the following proportion:
$$\frac{\text{smaller triangle side}}{\text{larger triangle side}} = \frac{\text{corresponding smaller triangle side}}{\text{corresponding larger triangle side}}$$
Using the given values, we have:
$$\frac{9 \text{ cm}}{24.3 \text{ cm}} = \frac{6 \text{ cm}}{x \text{ cm}}$$

4. Solve for x: To solve for $x$, we can cross-multiply:
$$9 \times x = 24.3 \times 6$$
$$9x = 145.8$$
Now, divide both sides by 9:
$$x = \frac{145.8}{9}$$
$$x = 16.2$$

1. State the final answer: The value of $x$ is 16.2 cm.