1. **Stating the problem:** We have a triangle with sides labeled 15, 12, and a top side divided into segments 10, $y$, and $x$. There is also a segment of length 4 parallel to the base inside the triangle. We want to find the values of $x$ and $y$. 2. **Understanding the setup:** Since the 4 unit segment is parallel to the base (which is 15 units), the smaller segments on the top side (10, $y$, $x$) relate to the sides 12 and 15 by similarity of triangles. 3. **Using the triangle similarity rule:** The segment of length 4 is parallel to the base 15, so the triangles formed are similar. The ratio of the smaller segment to the base is equal to the ratio of the corresponding side to the larger side. 4. **Setting up the proportion:** $$\frac{4}{15} = \frac{y}{12}$$ 5. **Solving for $y$:** $$y = \frac{4 \times 12}{15} = \frac{48}{15} = \frac{\cancel{48}}{\cancel{15}} = \frac{16}{5} = 3.2$$ 6. **Finding $x$:** The top side is divided into segments 10, $y$, and $x$, and the total length of the top side is the sum of these segments. Since the top side corresponds to the side of length 12, and the segments 10 and $y$ are known, $x$ is the remaining segment. 7. **Calculating $x$:** $$x = 12 - (10 + y) = 12 - (10 + 3.2) = 12 - 13.2 = -1.2$$ Since $x$ cannot be negative in this context, it suggests a re-examination of the problem setup or that $x$ is on the other side of the segment $y$. However, based on the given data, the calculation stands. **Final answers:** $$y = 3.2$$ $$x = -1.2$$