To analyze Julia's work, we first need to clarify the concept of similar triangles. The ratios of the corresponding sides of similar triangles are equal. In this case, we have:
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For the first triangle, the rise is 3 and the run is 10. Therefore, the ratio of rise to run is: \[ \frac{3}{10} \]
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For the second triangle, the rise is given as 7.5, and we need to find the corresponding run (let's call it \( x \)) so that the triangles are similar. This means we can set up the proportion as: \[ \frac{3}{10} = \frac{7.5}{x} \]
Now, simplifying Julia’s work step by step:
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The proportion can be set up correctly: \[ \frac{3}{10} = \frac{7.5}{x} \]
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Julia incorrectly wrote: \[ 3/10 = 7/5 / x \] Instead of using 7/5, it should have been 7.5, which is her rise.
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To solve for \( x \), we can cross multiply: \[ 3x = 10 \cdot 7.5 \] \[ 3x = 75 \] \[ x = \frac{75}{3} = 25 \]
Now analyzing Julia's steps:
- Her proportion was constructed incorrectly in step 1 since she used \( 7/5 \) instead of \( 7.5 \).
- However, her arithmetic in the subsequent steps was correctly executed, leading to finding \( x = 25 \).
To answer the question:
No, her proportion does not have the corresponding measures from corresponding figures in the same positions in the ratios.
Therefore, her mistake was in setting up the proportion incorrectly with incorrect corresponding values.
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