Opal has already proved that △ ADF ~ △ ABC. They are now attempting to prove that DF= 1/2 BC and DF||BC. Critique their reasoning. Which option should fill in the blank? PROOF: Because corresponding sides of 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|| BC. (1 point)

Question

A. DF over BC = 1/2
B. AD = DB.
C. BC over DF = 2.
D. DF = BC.

Answer 1

To critique Opal's reasoning, let's break down their proof steps and determine which option best fills in the blank.

1. Understanding Proportionality in Similar Triangles: Since △ ADF ~ △ ABC, it follows that the sides of these triangles are proportional. This means we can write the proportionality relationship as: $$\frac{DF}{BC}=\frac{AD}{AB}=\frac{AF}{AC}$$ This supports the statement that if $DF$ is the corresponding side to $BC$ in triangle $ABC$, then $DF$ must relate to $BC$ through a constant ratio.

2. The Given Statement in the Blank: The statement to fill in the blank should express this proportionality relation in a way that leads to concluding $DF=\frac{1}{2}BC$.

Considering the options:

• A. DF over BC = 1/2: This directly expresses the relationship that $DF$ is half of $BC$. This option fits perfectly into Opal's reasoning regarding the proportional sides of similar triangles, justifying the equation $DF\cdot 2=BC$.

• B. AD = DB: This statement is unrelated to the proportionality of $DF$ and $BC$.

• C. BC over DF = 2: While this is a valid statement (since it is mathematically equivalent to option A), it is not the format that Opal is using in the context of their proof. Opal is looking to express the proportion in terms of $DF$ being equal to half of $BC$.

• D. DF = BC: This is incorrect because that does not hold true if $DF$ is only half of $BC$ in the context of the similar triangles.

The most logical and direct option to fill in the blank that supports the conclusion that $DF=\frac{1}{2}BC$ is:

A. DF over BC = 1/2.

This correctly summarizes the relationship derived from the similarity of the triangles.