Question

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
(there is a triangle diagram on the right with vertices m, n, p (m---n---p on the base) and q at the top, with r on qm, connecting r to n.)

Explanation:

Step1: Set up ratio for Q3

We use the Triangle Proportionality Theorem (if a line divides two sides of a triangle proportionally, it is parallel to the third side). Calculate the ratios of the divided segments:
$\frac{PN}{PM} = \frac{6}{18} = \frac{1}{3}$
$\frac{RM}{QM} = \frac{16}{24} = \frac{2}{3}$

Step2: Compare ratios for Q3

Check if the ratios are equal:
$\frac{1}{3}
eq \frac{2}{3}$

Step3: Set up ratio for Q4

Let $PN = x$, so $PM = 4x$. Calculate the ratio of the divided segments on $MP$:
$\frac{PN}{PM} = \frac{x}{4x} = \frac{1}{4}$
Calculate the ratio on $MQ$:
$\frac{RM}{QM} = \frac{21}{31}$

Step4: Compare ratios for Q4

Check if the ratios are equal:
$\frac{1}{4}
eq \frac{21}{31}$

Answer:

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

eq \frac{RM}{QM}$.

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

eq \frac{RM}{QM}$.