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Opal has already proved that ∆ADF ~ ∆ABC. They are now attempting to prove that DF = 1/2 BC and DF ll BC. Critique their reasoning. Which option should fill in the blank?

PROOF: Because corresponding sides of the similar triangles are proportional, ________. Multiply both sides of the equation by segment BC to show that DF = 1/2 BC. Because corresponding angles of similar triangles are congruent, ∠ADF ≅ ∠ABC. If corresponding angles are congruent, then lines are parallel, so DF ll BC.

a. DF/BC = 1/2
b. BC/DF = 2
c. DF = BC
d. AD = DB

1 answer

To critique Opal's reasoning, we need to analyze their argument step by step.

  1. They state that ∆ADF ~ ∆ABC, which implies that corresponding sides of the triangles are proportional.

  2. They attempt to apply this fact in the proof. The blank in their reasoning appears to refer to the proportional relationship that results from the similarity of the triangles.

Given the similarity, we know the following proportion holds for the corresponding sides: \[ \frac{DF}{BC} = \frac{AD}{AB} = \frac{AF}{AC} \]

  1. They want to show that \(DF = \frac{1}{2} BC\). This would imply that: \[ \frac{DF}{BC} = \frac{1}{2} \]

Now, evaluating the options you provided:

  • a. DF/BC = 1/2: This option directly states the relationship that they seem to want to prove, which will allow them to conclude \(DF = \frac{1}{2} BC\).

  • b. BC/DF = 2: This is equivalent to saying \(DF = \frac{1}{2} BC\), but it does not directly state the proportionality as required in the blank.

  • c. DF = BC: This contradicts their goal, as they want to demonstrate that \(DF\) is half of \(BC\).

  • d. AD = DB: This does not relate directly to the proof regarding the lengths DF and BC.

The most appropriate option to fill the blank is a. DF/BC = 1/2 because it correctly captures the proportional relationship derived from the similarity of the triangles, allowing them to conclude that \(DF = \frac{1}{2} BC\).

Therefore, the critique reveals that Opal's proof is on the right track, but it needs the precise formulation of the similar triangles' proportion to clarify the reasoning. The completion of the proof hinges on correctly stating this proportionality.

Final Answer: a. DF/BC = 1/2.