Jenny has a triangular flag that has two sides 18 inches long and one side 6 inches long. She is making another triangular flag with angles equal in measure to the angles of the flag she has. If the new flag will have two sides 21 inches long, what will be the length of the third side?
The length of the third side will be 7 inches.
Using the properties of similar triangles, the new flag Jenny is making will maintain the same angle measures as the original flag. The sides of the new flag are scaled versions of the original sides, and by using the ratio of the corresponding sides, we can determine the length of the third side.
This choice is correct because the original triangle has a side length ratio of 18:6 (or 3:1) for the side lengths. If the two sides of the new flag are 21 inches, the corresponding side can be calculated using the ratio: \( \frac{21}{18} = \frac{x}{6} \), leading to \( x = 7 \) inches.
This choice is incorrect as it does not maintain the proportionality established by the original triangle's dimensions. Using the scaling factor derived from the sides of 21 and 18 inches would not yield a third side of 8 inches.
This option is also incorrect because it does not fit within the proportional scaling derived from the relationship between the sides of the original triangle and the sides of the new one. The calculations based on ratios do not support this length.
This choice is incorrect as well since it fails to adhere to the proportional relationship necessary for similar triangles. The calculated third side must align with the established ratios rather than being an arbitrary measurement.
By applying the principles of similar triangles, we determined that the length of the third side of Jenny's new flag is 7 inches. This conclusion results from maintaining the proportional relationships between the side lengths of the original flag and the new flag, ensuring that all angles and side lengths correlate appropriately.
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