Question

Use the image to answer the question.

A triangle is marked clockwise from the lower left vertex as upper A upper B upper C. Point upper D is marked at the midpoint of side upper A upper B. Point upper F is marked at the midpoint of side upper A upper C.

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

Choose the last option: DF/BC = 1/2.

Better justification in full: from similarity DF/BC = AD/AB; since D is the midpoint AD/AB = 1/2, so DF/BC = 1/2 and multiplying by BC gives DF = (1/2)BC. The angle correspondence then gives ∠ADF ≅ ∠ABC, so DF ∥ BC.