Question

for which segment lengths is \\(\overline{qs}\\) parallel to \\(\overline{mn}\\)?

image of a triangle with points q, r, s; m on qr, n on rs

select parallel or not parallel for each set of given information.

segment lengths\tparallel\tnot parallel

Explanation:

To determine when \(\overline{QS}\) is parallel to \(\overline{MN}\), 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 a line divides two sides of a triangle proportionally, it is parallel to the third side.

Step 1: Recall the Theorem

For \(\triangle QRS\) (assuming \(M\) is on \(QR\) and \(N\) is on \(RS\)), \(\overline{MN} \parallel \overline{QS}\) if and only if:
\[
\frac{QM}{MR} = \frac{SN}{NR}
\]

Step 2: Analyze Each Set (Example)

Suppose we have:

• Set 1: \(QM = 2\), \(MR = 4\), \(SN = 3\), \(NR = 6\)

Check \(\frac{QM}{MR} = \frac{2}{4} = \frac{1}{2}\) and \(\frac{SN}{NR} = \frac{3}{6} = \frac{1}{2}\).
Since \(\frac{QM}{MR} = \frac{SN}{NR}\), \(\overline{MN} \parallel \overline{QS}\) (Parallel).

• Set 2: \(QM = 3\), \(MR = 5\), \(SN = 4\), \(NR = 7\)

Check \(\frac{QM}{MR} = \frac{3}{5} = 0.6\) and \(\frac{SN}{NR} = \frac{4}{7} \approx 0.571\).
Since \(\frac{QM}{MR}
eq \frac{SN}{NR}\), \(\overline{MN}\) is Not parallel to \(\overline{QS}\).

General Process

For any set of segment lengths:

1. Calculate \(\frac{QM}{MR}\) (ratio of segments on \(QR\)).
2. Calculate \(\frac{SN}{NR}\) (ratio of segments on \(RS\)).
3. If the ratios are equal, \(\overline{MN} \parallel \overline{QS}\) (Parallel); otherwise, Not parallel.

Example Table (Fill with Given Lengths)

Segment Lengths (e.g., \(QM, MR, SN, NR\))	Parallel	Not Parallel
\(QM=3, MR=5, SN=4, NR=7\)	❌	✔️

To apply this to your specific problem, substitute the given segment lengths into the ratio formula and compare. If the ratios match, mark “Parallel”; otherwise, “Not parallel.”

Answer:

To determine when \(\overline{QS}\) is parallel to \(\overline{MN}\), 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 a line divides two sides of a triangle proportionally, it is parallel to the third side.

Step 1: Recall the Theorem

For \(\triangle QRS\) (assuming \(M\) is on \(QR\) and \(N\) is on \(RS\)), \(\overline{MN} \parallel \overline{QS}\) if and only if:
\[
\frac{QM}{MR} = \frac{SN}{NR}
\]

Step 2: Analyze Each Set (Example)

Suppose we have:

• Set 1: \(QM = 2\), \(MR = 4\), \(SN = 3\), \(NR = 6\)

Check \(\frac{QM}{MR} = \frac{2}{4} = \frac{1}{2}\) and \(\frac{SN}{NR} = \frac{3}{6} = \frac{1}{2}\).
Since \(\frac{QM}{MR} = \frac{SN}{NR}\), \(\overline{MN} \parallel \overline{QS}\) (Parallel).

• Set 2: \(QM = 3\), \(MR = 5\), \(SN = 4\), \(NR = 7\)

Check \(\frac{QM}{MR} = \frac{3}{5} = 0.6\) and \(\frac{SN}{NR} = \frac{4}{7} \approx 0.571\).
Since \(\frac{QM}{MR}
eq \frac{SN}{NR}\), \(\overline{MN}\) is Not parallel to \(\overline{QS}\).

General Process

For any set of segment lengths:

1. Calculate \(\frac{QM}{MR}\) (ratio of segments on \(QR\)).
2. Calculate \(\frac{SN}{NR}\) (ratio of segments on \(RS\)).
3. If the ratios are equal, \(\overline{MN} \parallel \overline{QS}\) (Parallel); otherwise, Not parallel.

Example Table (Fill with Given Lengths)

Segment Lengths (e.g., \(QM, MR, SN, NR\))	Parallel	Not Parallel
\(QM=3, MR=5, SN=4, NR=7\)	❌	✔️

To apply this to your specific problem, substitute the given segment lengths into the ratio formula and compare. If the ratios match, mark “Parallel”; otherwise, “Not parallel.”