Question

given the figure, and $overline{mn}paralleloverline{qr}$, what can be concluded? mp = 2·qm $\frac{pm}{pq}=\frac{pn}{pr}$ $\frac{pm}{nr}=\frac{pn}{mq}$ mn = qr

Explanation:

Step1: Recall similarity - triangle property

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. In \(\triangle PQR\), since \(MN\parallel QR\), by the basic - proportionality theorem (Thales' theorem), we have \(\frac{PM}{PQ}=\frac{PN}{PR}\).

Step2: Analyze each option

• Option \(MP = 2\cdot QM\): There is no information given to support this ratio.
• Option \(\frac{PM}{PQ}=\frac{PN}{PR}\): This is correct according to the basic - proportionality theorem for similar triangles formed by the parallel line \(MN\) and \(\triangle PQR\).
• Option \(\frac{PM}{NR}=\frac{PN}{MQ}\): This is not a valid proportion based on the parallel - line and triangle relationship.
• Option \(MN = QR\): Just because \(MN\parallel QR\) does not mean they are equal in length.

Answer:

\(\frac{PM}{PQ}=\frac{PN}{PR}\)