To determine the correct ratio of the triangles, we can use the properties of similar triangles.
In similar triangles, the ratios of the lengths of corresponding sides are equal. We can designate the following:
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Triangle ABC (the smaller triangle):
- \( AB \) is the side adjacent to angle A
- \( BC \) is the side opposite angle A
- \( AC \) is the hypotenuse
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Triangle DEF (the larger triangle):
- \( DE \) is the side adjacent to angle D
- \( EF \) is the side opposite angle D
- \( DF \) is the hypotenuse
Since the two triangles are right triangles, we want to compare the ratios of their sides.
Let's break down the options:
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ABBC = ACDF = DEEF: This doesn't make sense because it seems to misplace terms in relationships between side lengths.
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\( \frac{AB}{EF} = \frac{BC}{DF} = \frac{AC}{DE} \): This form could correspond to a similarity ratio, as it is taking ratios of segments that should compare a side of the smaller triangle to a side of the larger triangle.
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ABDE = BCEF = ACDF: Similar to the first option, this does not correctly represent side ratios.
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ABDF = BCEF = ACDE: While tempting, again doesn't follow the usual use of side ratios.
Among these, the second option:
\( \frac{AB}{EF} = \frac{BC}{DF} = \frac{AC}{DE} \) seems the most plausible.
This fits the definition of similar triangles and ratios of corresponding sides—they maintain the same proportions due to the similar triangle property.
In conclusion, the correct choice is the second one:
Start Fraction upper A upper B over upper E upper F End Fraction equals Start Fraction upper B upper C over upper D upper F End Fraction equals Start Fraction upper A upper C over upper D upper E End Fraction.
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