To determine if Julia correctly calculated the length of the run for the second triangle, we need to analyze the ratios of the corresponding sides of similar triangles.
The first triangle has:
- Rise = 3
- Run = 10
This gives the ratio of rise to run for the first triangle:
\[ \text{Ratio}_1 = \frac{3}{10} \]
For the second triangle:
- Rise = 7.5
- Let the run be \( x \).
Since the triangles are similar, their corresponding sides will maintain the same ratio:
\[ \frac{3}{10} = \frac{7.5}{x} \]
To solve for \( x \), we can cross-multiply:
\[ 3x = 10 \times 7.5 \]
Calculating the right side:
\[ 10 \times 7.5 = 75 \]
So we have:
\[ 3x = 75 \]
Now, divide both sides by 3:
\[ x = \frac{75}{3} = 25 \]
Therefore, the run for the second triangle should be 25 in order for the two triangles to be similar.
Now, let's evaluate the response options:
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No, she needed to multiply 7.5 to both sides. - This is incorrect; it's not about multiplying both sides but rather maintaining the ratio.
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Yes, she is correct. - This would only be true if Julia had calculated the run correctly, which we've found she did not.
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No, her proportion does not have the corresponding measures from corresponding figures in the same positions in the ratios. - This could be a valid point if she mistakenly set up the ratio incorrectly, but the main error lies in her calculation.
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No, she needed to have the product of the numerators equal to the product of the denominators. - This describes the correct process for solving proportions but doesn't address an actual mistake in her work itself.
The main mistake could have been in not setting up the proportion correctly, as a specific calculation error isn't pointed out.
Julia's work was incorrect, and the correct answer is that she made an error in not calculating the run of 25 properly based on the ratio.
In conclusion, the best response to the question would be: No, her proportion does not have the corresponding measures from corresponding figures in the same positions in the ratios.
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