Use the image to answer the question.

Question

There are two right triangles of different sizes. A smaller triangle is on the upper left and is labeled upper A, upper B, upper C. A larger triangle on the right is labeled upper D, upper E, upper F.

Determine which of the following ratios is correct.

(1 point)
Responses

ABEF=BCDF=ACDE
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

ABDE=BCEF=ACDF
Start Fraction upper A upper B over upper D upper E End Fraction equals Start Fraction upper B upper C over upper E upper F End Fraction equals Start Fraction upper A upper C over upper D upper F End Fraction

ABDF=BCEF=ACDE
Start Fraction upper A upper B over upper D upper F End Fraction equals Start Fraction upper B upper C over upper E upper F End Fraction equals Start Fraction upper A upper C over upper D upper E End Fraction

ABBC=ACDF=DEEF
Start Fraction upper A upper B over upper B upper C End Fraction equals Start Fraction upper A upper C over upper D upper F End Fraction equals Start Fraction upper D upper E over upper E upper F End Fraction

Answer 1

The question pertains to the relationships between the sides of two similar right triangles, the smaller triangle (ABC) and the larger triangle (DEF).

Since the description indicates that they are similar triangles, we can use the properties of similar triangles to determine the ratios of their corresponding sides.

In similar triangles, the ratios of the lengths of corresponding sides are equal. Thus if we denote the lengths of sides $AB$, $BC$, and $AC$ for triangle $ABC$ and lengths $DE$, $EF$, and $DF$ for triangle $DEF$, the correct expression for the ratios of the sides must reflect the equality based on their corresponding sides.

The correct option appears to use the notation and statements that identify the equal ratios. The best fit based on this understanding is likely similar triangles and is consistent with the relationships given.

Let’s analyze the options:

1. Option 1 states:
   $\frac{AB}{EF}=\frac{BC}{DF}=\frac{AC}{DE}$
   This is a valid statement indicating the ratios of the sides based on similarity.

2. Option 2 states:
   $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$
   This does not hold true as it mixes corresponding sides inaccurately.

3. Option 3 states:
   $\frac{AB}{DF}=\frac{BC}{EF}=\frac{AC}{DE}$
   Similar to option 2, this one also is inaccurate.

4. Option 4 states:
   $\frac{AB}{BC}=\frac{AC}{EF}=\frac{DE}{EF}$
   Again, this does not relate to corresponding sides correctly.

Thus, the first option maintains the proportions of the similar triangles correctly and is the most accurate representation based on the properties of proportional sides in similar triangles:

Final Answer: 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