To complete the proof, we need to select an option that properly explains the relationship between the sides of the similar triangles. Since Opal has already established that triangle ADF is similar to triangle ABC (△ADF∼△ABC), it means that the corresponding sides of these triangles are proportional.
Given that the problem states Opal is trying to prove that \( DF = \frac{1}{2} BC \), we can determine the correct option to fill in the blank based on proportionality.
The relevant statement to fill the blank could be:
"Because corresponding sides of similar triangles are proportional, \( \frac{DF}{BC} = \frac{AD}{AB} \) (or another ratio based on the given triangles), which allows us to multiply both sides by segment BC to show that \( DF = \frac{1}{2} BC \)."
Out of the responses listed:
- DF = 12BC
- DFBC = 12 Start Fraction upper D upper F over upper B upper C End Fraction equals Start Fraction 1 over 2 End Fraction
- AD = DB
- BCDF = 2 Start Fraction upper B upper C over upper D upper F End Fraction equals 2
The most appropriate choice would likely be one that mentions the proportional relationship such as:
"DF and BC are in a ratio of 1:2."
However, none of the provided responses exactly match this phrasing. If we had the option to choose the closest related expression concerning the proportionality concept, it would be something like "Start Fraction upper D upper F over upper B upper C End Fraction equals Start Fraction 1 over 2 End Fraction".
In summary, we are looking for an expression that conveys that the sides are proportional and can be simplified to show \( DF = \frac{1}{2} BC \). If "Start Fraction upper D upper F over upper B upper C End Fraction equals Start Fraction 1 over 2 End Fraction" is supposed to imply that relationship correctly, that would be the best fit.
Let me know if you have additional context or need further assistance!
1 answer