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Opal has already proved that △ADF∼△ABC . They are now attempting to prove that DF=12BC 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=12BC . Because corresponding angles of similar triangles are congruent, ∠ADF≅∠ABC . If corresponding angles are congruent, then lines are parallel, so DF¯¯¯¯¯¯¯¯∥BC¯¯¯¯¯¯¯¯ . (1 point) Responses AD=DB upper A upper D equals upper D upper B DFBC=12 Start Fraction upper D upper F over upper B upper C End Fraction equals Start Fraction 1 over 2 End Fraction DF=BC upper D upper F equals upper B upper C BCDF=2 Start Fraction upper B upper C over upper D upper F End Fraction equals 2 Skip to navigation

1 answer

To critique Opal's reasoning, let's focus on the claim being made: "Because corresponding sides of similar triangles are proportional, ________________." In the context of similar triangles, the statement must express a relationship that leads to the conclusion about the sides.

The most appropriate option to fill in the blank would be:

"DF/BC = 1/2."

Here’s a breakdown of the reasoning:

  1. Since triangles \( \triangle ADF \sim \triangle ABC \), it follows that corresponding sides are proportional. This means that: \[ \frac{DF}{BC} = k \text{ (for some constant } k\text{)} \]
  2. If they suspect \( DF \) is exactly half the length of \( BC \), then \( k \) would be \( \frac{1}{2} \), which gives: \[ \frac{DF}{BC} = \frac{1}{2} \]
  3. Thus, multiplying both sides by \( BC \) leads to: \[ DF = \frac{1}{2} BC \]

So in conclusion, the correct option to fill in their blank is: "DF/BC = 1/2."

This accurately sets up the proportional relationship needed in their proof. The other options either do not express the required relationship or are phrased incorrectly for the context of similar triangles.