Question

A line connects midpoints upper D and upper F. Segment upper A upper D and segment upper D upper B are marked with single congruent tick marks. Segment upper A upper F, and segment upper F upper C are marked with double congruent tick marks.


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

AD=DB

upper A upper D equals upper D upper B

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

DFBC=12

Answer 1

The blank should state the proportionality of corresponding sides: DF/BC = AD/AB. Using AD = DB (D is midpoint) gives AD/AB = 1/2, so DF/BC = 1/2 and multiplying by BC yields DF = (1/2)BC.

Of the given choices, pick the one that means DF/BC = 1/2 (the option shown as "DFBC=12").