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=AP
upper Q upper N equals upper A upper P

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

To analyze Jamal's findings and draw a conclusion about the length of \( QN \), let’s summarize the given information step by step:

1. Midsegment Relationship:

   • Jamal has proven that \( MN = \frac{1}{2} AC \) and also established that \( MN \parallel AC \) using the Triangle Midsegment Theorem. This indicates that the midsegment (MN) is half the length of the side (AC) of the triangle.

2. Similarity of Triangles:

   • Jamal has established that \( \triangle QBN \sim \triangle PBC \) via the AAA Similarity Theorem. From similar triangles, corresponding side lengths are proportional.

Since \( MN \) is parallel to \( AC \) and since \( QN \) is a segment associated with the triangles formed with the points, we can use the similarity of the triangles \( QBN \) and \( PBC \) to help relate \( QN \) to sides or segments we know.

Given:

• \( MN \) being the midsegment and thus \( MN = \frac{1}{2} AC \),
• The triangles \( QBN \) and \( PBC \) being similar, we can express the length of \( QN \) in relation to the segments of triangle \( ABC \).

Since \( MN \) is parallel to \( AC \), it divides triangle \( ABC \) into a smaller triangle and a trapezoid, preserving the ratio of corresponding sides from the similarity of triangles \( QBN \) and \( PBC \).

Looking at the options available, we recognize:

• From the triangle similarity \( \triangle QBN \sim \triangle PBC \), the ratios of their sides imply that:

  \[ \frac{QN}{PC} = \frac{1}{2} \]

This means that \( QN \) is half the length of \( AC \):

If \( AC \) (directly corresponding equivalent of \( PC \) in the larger triangle) is defined as a baseline, then:

In relation to \( AC \):

Thus, \( QN \) can be concluded as:

QN = 1/2 AC

Accordingly, the correct response is:

QN = \frac{1}{2} AC