Question

For which segment lengths is QS¯¯¯¯¯ parallel to MN¯¯¯¯¯¯¯ ? A triangle with vertices labeled as Q, R, S. Side Q S is base. Sides R Q and R S contain midpoints M and N, respectively. A line segment is drawn from M to N. Select Parallel or Not parallel for each set of given information. Segment lengths Parallel Not parallel QM=2, MR=4, SN=3, NR=5 QM=2, MR=5, SN=6, NR=15 QM=2, MR=8, SN=3, NR=12

Answer 1

QM=2 MR=4 SN=3 NR=3) NOT Parallel
QM=2 MR=5 SN=6 NR=15) PARALLEL
QM=2 MR=8 SN=3 NR=12) PARALLEL 
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Answer 2

Answer:

Thus, (1) is non parallel set and (2) and (3) are parallel set.

Step-by-step explanation:

Consider a triangle with vertices labeled as Q, R, S. Side QS is base. Sides RQ and RS contain midpoints M and N, respectively. A line segment is drawn from M to N as shown in figure below.  

For MN and QS to be parallel the ratio of line segment must be in same ratio, that is $\frac{RM}{RQ}=\frac{RN}{RS}$.

Check for the given options,

1) Given : QM=2, MR=4, SN=3, NR=5

Find the values in the ratio,

$\frac{RM}{RQ}=\frac{RM}{RM+MQ}=\frac{4}{6}=\frac{2}{3}$

$\frac{RN}{RS}=\frac{RN}{RN+NS}=\frac{5}{8}=\frac{5}{8}$

Thus, $\frac{RM}{RQ}\neq \frac{RN}{RS}$

Thus, (1) is non parallel set.

2) Given : QM=2, MR=5, SN=6, NR=15  

Find the values in the ratio,

$\frac{RM}{RQ}=\frac{RM}{RM+MQ}=\frac{5}{7}$

$\frac{RN}{RS}=\frac{RN}{RN+NS}=\frac{15}{21}=\frac{5}{7}$

Thus, $\frac{RM}{RQ}=\frac{RN}{RS}$

Thus, (2) is parallel set.

3) Given: QM=2, MR=8, SN=3, NR=12

Find the values in the ratio,

$\frac{RM}{RQ}=\frac{RM}{RM+MQ}=\frac{8}{10}=\frac{4}{5}$

$\frac{RN}{RS}=\frac{RN}{RN+NS}=\frac{12}{15}=\frac{4}{5}$

Thus, $\frac{RM}{RQ}=\frac{RN}{RS}$

Thus, (3) is parallel set.