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
△
𝐴
𝐷
𝐹
∼
△
𝐴
𝐵
𝐶
. They are now attempting to prove that DF=12BC
𝐷
𝐹
=
1
2
𝐵
𝐶
and DF¯¯¯¯¯¯¯¯∥BC¯¯¯¯¯¯¯¯
𝐷
𝐹
¯
∥
𝐵
𝐶
¯
. 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
𝐵
𝐶
to show that DF=12BC
𝐷
𝐹
=
1
2
𝐵
𝐶
. Because corresponding angles of similar triangles are congruent, ∠ADF≅∠ABC
∠
𝐴
𝐷
𝐹
≅
∠
𝐴
𝐵
𝐶
. If corresponding angles are congruent, then lines are parallel, so DF¯¯¯¯¯¯¯¯∥BC¯¯¯¯¯¯¯¯
𝐷
𝐹
¯
∥
𝐵
𝐶
¯
.

DFBC=12
𝐷
𝐹
𝐵
𝐶
=
1
2

Start Fraction upper D upper F over upper B upper C End Fraction equals Start Fraction 1 over 2 End Fraction

BCDF=2
𝐵
𝐶
𝐷
𝐹
=
2

Start Fraction upper B upper C over upper D upper F End Fraction equals 2

AD=DB
𝐴
𝐷
=
𝐷
𝐵

upper A upper D equals upper D upper B

DF=BC

Answer 1

Choose "DF/BC = 1/2."

Critique / correct reasoning (concise):
- From △ADF ∼ △ABC we get corresponding-side proportions: DF/BC = AD/AB = AF/AC.
- Since D and F are midpoints, AD = DB so AD/AB = 1/2 (and AF/AC = 1/2). Hence DF/BC = 1/2.
- Multiplying both sides by BC gives DF = (1/2)BC.
- Corresponding angles of the similar triangles are congruent, so ∠ADF ≅ ∠ABC, hence DF ∥ BC.

Note: The blank "AD = DB" is true (given by midpoints) but does not directly follow from similarity; the blank should be the proportional relation (DF/BC = 1/2, or more generally DF/BC = AD/AB).