In this image, we are given two triangles that appear to be similar, and the goal is to find the values of $x$ and $y$. Since the triangles are similar, the sides of the triangles will be proportional to each other.

Step 1: Set up the proportion

The smaller triangle has sides labeled 2, 4, and 5, while the larger triangle has sides labeled $x$, $y$, and 12. To find $x$ and $y$, we will use the fact that corresponding sides are proportional.

Let's match the corresponding sides:

• The side with length 2 corresponds to the side with length $x$,
• The side with length 5 corresponds to the side with length 12,
• The side with length 4 corresponds to the side with length $y$.

So, we can set up the following proportions: $\frac{x}{2} = \frac{12}{5}$ $\frac{y}{4} = \frac{12}{5}$

Step 2: Solve for $x$

To solve for $x$, multiply both sides of the first proportion by 2: $x = 2 \times \frac{12}{5} = \frac{24}{5} = 4.8$ Thus, $x \approx 4.8$, which rounds to 5, matching one of the answer choices.

Step 3: Solve for $y$

To solve for $y$, multiply both sides of the second proportion by 4: $y = 4 \times \frac{12}{5} = \frac{48}{5} = 9.6$ Thus, $y \approx 9.6$, which rounds to 10, also matching one of the answer choices.

Step 4: Conclusion

Based on the calculations, $x \approx 5$ and $y \approx 10$. However, none of the given answer options directly match this, so there might be a rounding issue, or the problem could involve more precise values. Please review the provided answer choices again.

Do you need further clarifications, or is there any other question you'd like to discuss regarding this problem?

5 Related Questions:

1. How do we determine if triangles are similar?
2. What is the importance of corresponding sides in solving for unknowns in similar triangles?
3. Can you solve for unknown angles in similar triangles using proportions?
4. How do scaling factors affect the area of similar triangles?
5. Can this method be applied to non-right triangles as well?

Tip:

When working with similar triangles, always match corresponding sides and angles carefully to avoid errors in setting up the proportions.