Question

for which segment lengths is \\(\overline{ln}\\) parallel to \\(\overline{xy}\\)?
(image of a triangle with points m, l, n, and x on ml, y on mn)
select parallel or not parallel for each set of given information.

segment lengths	parallel	not parallel
\\(lx = 2, xm = 6, ny = 3, ym = 9\\)	\\(\circ\\)	\\(\circ\\)

Explanation:

To determine if \(\overline{LN}\) is parallel to \(\overline{XY}\), we use the Basic Proportionality Theorem (Thales' theorem), which states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. Conversely, if the line divides the sides proportionally, then it is parallel to the third side.

For a triangle \( \triangle MLN \) with a line \( XY \) intersecting \( ML \) at \( X \) and \( MN \) at \( Y \), the theorem gives the proportion:
\[
\frac{LX}{XM} = \frac{NY}{YM}
\]
if and only if \( XY \parallel LN \).

Step 1: Analyze the first set of segment lengths (\( LX = 2 \), \( XM = 7 \), \( NY = 4 \), \( YM = 14 \))

Calculate the ratios \(\frac{LX}{XM}\) and \(\frac{NY}{YM}\):
\[
\frac{LX}{XM} = \frac{2}{7}
\]
\[
\frac{NY}{YM} = \frac{4}{14} = \frac{2}{7}
\]
Since \(\frac{LX}{XM} = \frac{NY}{YM}\), by the Basic Proportionality Theorem, \( XY \parallel LN \) for this set.

Step 2: Analyze the second set of segment lengths (\( LX = 2 \), \( XM = 6 \), \( NY = 3 \), \( YM = 9 \))

Calculate the ratios \(\frac{LX}{XM}\) and \(\frac{NY}{YM}\):
\[
\frac{LX}{XM} = \frac{2}{6} = \frac{1}{3}
\]
\[
\frac{NY}{YM} = \frac{3}{9} = \frac{1}{3}
\]
Since \(\frac{LX}{XM} = \frac{NY}{YM}\), by the Basic Proportionality Theorem, \( XY \parallel LN \) for this set as well.

Answer:

For \( LX = 2 \), \( XM = 7 \), \( NY = 4 \), \( YM = 14 \): Parallel
For \( LX = 2 \), \( XM = 6 \), \( NY = 3 \), \( YM = 9 \): Parallel