Question

Use the image to answer the question.

Triangle upper A upper B upper C has two line segments inside the triangle. Four points are also plotted, three on the triangle’s sides and one inside of the triangle.

Jamal is attempting to find out the relationship between QN
and AC
. He has already proven the following:

MN=12AC
and MN¯¯¯¯¯¯¯¯¯¯∥AC¯¯¯¯¯¯¯¯
by applying the Triangle Midsegment Theorem.
△QBN∼△PBC
by applying the AAA Similarity Theorem.
Using what he has proven, what can Jamal conclude about the length of QN¯¯¯¯¯¯¯¯
?

(1 point)
Responses

QN=AP
upper Q upper N equals upper A upper P

QN=12AC
upper Q upper N equals Start Fraction 1 over 2 End Fraction upper A upper C

QN=14AC
upper Q upper N equals Start Fraction 1 over 4 End Fraction upper A upper C

QN=PC
upper Q upper N equals upper P upper C
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Answer 1

QN = 1/4 AC.

Reason: MN is a midsegment so N is the midpoint of BC, hence BN = (1/2)BC. From △QBN ∼ △PBC we get BN/BC = QN/PC, so QN = (1/2)PC. P is the midpoint of AC, so PC = (1/2)AC. Therefore QN = (1/2)(1/2)AC = 1/4 AC.