To determine the correct proportion to show that the slope of line segment \( XY \) (denoted as \( XX \) which could be a typo) is equal to the slope of line segment \( ZU \) in similar right triangles \( XYZ \) and \( ZVU \), we can use the formula for the slope of a line segment, which is:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we are given options that seem to relate to slopes of two line segments, but the values are not contextualized with specific points from the triangles. The proportions provided seem to contain expressions for the slope, so we need to analyze each option based on the form of the slope:
- \( A: \frac{1 - 5}{-7 - 3} \)
- \( B: \frac{-1 + 5}{7 - 3} \)
- \( C: \frac{-1 - 5}{7 - 3} \)
- \( D: \frac{-1 - 5}{7 + 3} \)
To solve it, let's simplify the expressions for the slopes:
Option A: \[ \frac{1 - 5}{-7 - 3} = \frac{-4}{-10} = \frac{2}{5} \]
Option B: \[ \frac{-1 + 5}{7 - 3} = \frac{4}{4} = 1 \]
Option C: \[ \frac{-1 - 5}{7 - 3} = \frac{-6}{4} = -\frac{3}{2} \]
Option D: \[ \frac{-1 - 5}{7 + 3} = \frac{-6}{10} = -\frac{3}{5} \]
Now, since the triangles \( XYZ \) and \( ZVU \) are similar, their slopes should be proportional to each other. Without more specific information about the coordinates of points \( X, Y, Z, V, U \), we cannot definitively conclude which option is correct.
However, if you are looking for a proportion that could show similarity in slopes, generally we can set them equal as:
\[ \text{slope of } XY = \text{slope of } ZU \]
If you provide further details about the positions of the points X, Y, Z, V, U, or the context behind the slopes you are asking about, I could help you find a more precise conclusion. Otherwise, based on the slopes calculated, you can choose the appropriate proportion from what fits your requirements for similar triangles.
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