Question

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, ________________Blank. 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

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

AD=DB

upper A upper D equals upper D upper B

DFBC=12

Answer 1

Fill the blank with DF/BC = 1/2 (the option shown as "DFBC=12").

Critique: From ΔADF ∼ ΔABC we get the proportionality DF/BC = AD/AB. If it's given (or previously proved) that AD = DB, then AD/AB = 1/2, so DF/BC = 1/2 and multiplying both sides by BC gives DF = (1/2)BC. The angle argument (∠ADF ≅ ∠ABC ⇒ DF ∥ BC) is valid.