Question

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=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

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
Skip to navigation

Answer 1

Based on the information provided:

1. Jamal has proven that \( MN = \frac{1}{2} AC \) and that \( MN \parallel AC \) by using the Triangle Midsegment Theorem.
2. He has also established that \( \triangle QBN \sim \triangle PBC \) by the AAA Similarity Theorem.

Since \( MN \parallel AC \) and \( QN \) is a segment that corresponds to \( AC \) in the similar triangles \( \triangle QBN \) and \( \triangle PBC \), we can use the properties of similar triangles to find the ratio of their corresponding sides.

Because \( MN \) is the midsegment and is equal to half the length of \( AC \), this indicates that the corresponding side \( QN \) would also be half the length of the side it corresponds with in triangle \( PBC \).

Thus, if \( MN \) (which is half of \( AC \)) is parallel to \( AC \), then the length \( QN \) should also be similarly proportional to \( AC \).

From \( \triangle QBN \sim \triangle PBC \), since the similarity ratio is preserved, we conclude: \[ QN = \frac{1}{2} AC \]

So, the correct response is:

QN = \( \frac{1}{2} AC \).