Question

example 2
determine whether $overline{nr} \parallel overline{pq}$. justify your answer.

1. $pm = 18$, $pn = 6$, $qm = 24$, and $rm = 16$
2. $qm = 31$, $rm = 21$, and $pm = 4pn$

example 3

Explanation:

Step1: Check proportionality for Q3

We use the converse of the triangle proportionality theorem: if $\frac{PM}{PN}=\frac{QM}{RM}$, then $\overline{NR} \parallel \overline{PQ}$.
Calculate $\frac{PM}{PN}=\frac{18}{6}=3$, $\frac{QM}{RM}=\frac{24}{16}=1.5$

Step2: Conclude for Q3

Since $3
eq 1.5$, the sides are not proportional.

Step3: Define variables for Q4

Let $PN = x$, so $PM = 4x$. Then $MN = PM - PN = 4x - x = 3x$.

Step4: Check proportionality for Q4

Calculate $\frac{MN}{PN}=\frac{3x}{x}=3$, $\frac{RM}{QM}=\frac{21}{31}\approx0.677$

Step5: Conclude for Q4

Since $3
eq \frac{21}{31}$, the sides are not proportional.

Answer:

1. $\overline{NR}$ is not parallel to $\overline{PQ}$, because $\frac{PM}{PN}

eq \frac{QM}{RM}$.

1. $\overline{NR}$ is not parallel to $\overline{PQ}$, because $\frac{MN}{PN}

eq \frac{RM}{QM}$.