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=14AC upper Q upper N equals Start Fraction 1 over 4 End Fraction upper A upper C QN=12AC upper Q upper N equals Start Fraction 1 over 2 End Fraction upper A upper C QN=AP upper Q upper N equals upper A upper P QN=PC

Answer 1

To determine the relationship between \( QN \) and \( AC \), let's analyze the information provided.

1. Triangle Midsegment Theorem:

   • It states that a segment connecting the midpoints of two sides of a triangle is parallel to the third side and equal to half its length.
   • From your statement, \( MN = 12AC \) and \( MN \parallel AC \). This indicates that \( MN \) is not the midsegment in relation to \( AC \) under normal circumstances (it would be \( \frac{1}{2}AC \)); thus here it indicates that \( AC = \frac{1}{12}MN \).

2. AAA Similarity Theorem:

   • This theorem indicates that if two triangles are similar, the ratios of their corresponding sides are proportional.
   • Given that \( \triangle QBN \sim \triangle PBC \), we know the lengths of sides are proportional, which implies that: \[ \frac{QN}{AC} = \frac{QB}{PB} = \frac{BN}{BC} \]

Since \( MN \) is established as being related to \( AC \) and is parallel to it, we can conclude that since \( QN \) is a proportional part of the triangle's configuration, we can apply the previous conclusions.

Given that \( MN = 12AC \), and it is a factored relationship, we know this can directly influence our length comparison. Since triangles \( QBN \) and \( PBC \) must maintain the proportion of similar lengths, and we already have a multiple \( MN \) corresponding to \( AC \):

• Conclusion on \( QN \): If we analyze the potential options based on the ratios developed from the previous statements and the similarity, we can infer a ratio for \( QN \).

Thus, combining and simplifying the relations, we find that:

1. Given the proportionality \( QN = \frac{1}{2}AC \) would be a direct response inferred from \( AC \).

Thus, the best conclusion for Jamal regarding \( QN \) given the parameters would be:
\[ QN = \frac{1}{2}AC \] This matches the option presented: \( QN = \frac{1}{2}AC \).