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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

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

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

ABEF=BCDF=ACDE

1 answer

To determine the correct ratios involving the two right triangles, we know that similar right triangles have corresponding sides that are proportional. The ratios of the lengths of corresponding sides can be used to establish which option is correct.

Based on the standard naming convention for right triangles:

  • Triangle ABC (smaller triangle) has sides AB, BC, and AC.
  • Triangle DEF (larger triangle) has sides DE, EF, and DF.

If triangles ABC and DEF are similar, then the following ratios hold true:

  1. The ratio of the sides AB to DE corresponds to the ratio of sides BC to EF and also the ratio of sides AC to DF.

Using this reasoning, let's analyze each of the options:

  1. ABBC=ACDF=DEEF - This option does not show ratios and seems incorrect.

  2. Start Fraction AB over BC End Fraction = Start Fraction AC over DF End Fraction = Start Fraction DE over EF End Fraction - This appears incorrect as these do not represent the correct side ratios for similar triangles.

  3. ABDE=BCEF=ACDF - This also seems incorrect since it does not utilize side ratios correctly.

  4. Start Fraction AB over DE End Fraction = Start Fraction BC over EF End Fraction = Start Fraction AC over DF End Fraction - This representation correctly depicts the sides of the triangles, implying they are proportional.

  5. ABEF=BCDF=ACDE - This option does not represent the ratios correctly.

Thus, the best conclusion based on the concept of similar triangles and the ratios of their sides indicates that option 4 is the correct answer:

Start Fraction AB over DE End Fraction = Start Fraction BC over EF End Fraction = Start Fraction AC over DF End Fraction.