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

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

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
`(RM)/(RQ)=(RN)/(RS)`.

Check for the given options,

1) Given : QM=2, MR=4, SN=3, NR=5

Find the values in the ratio,

`(RM)/(RQ)=(RM)/(RM+MQ)=(4)/(6)=(2)/(3)`

`(RN)/(RS)=(RN)/(RN+NS)=(5)/(8)=(5)/(8)`

Thus,
`(RM)/(RQ)\\eq (RN)/(RS)`

Thus, (1) is non parallel set.

2) Given : QM=2, MR=5, SN=6, NR=15

Find the values in the ratio,

`(RM)/(RQ)=(RM)/(RM+MQ)=(5)/(7)`

`(RN)/(RS)=(RN)/(RN+NS)=(15)/(21)=(5)/(7)`

Thus,
`(RM)/(RQ)=(RN)/(RS)`

Thus, (2) is parallel set.

3) Given: QM=2, MR=8, SN=3, NR=12

Find the values in the ratio,

`(RM)/(RQ)=(RM)/(RM+MQ)=(8)/(10)=(4)/(5)`

`(RN)/(RS)=(RN)/(RN+NS)=(12)/(15)=(4)/(5)`

Thus,
`(RM)/(RQ)=(RN)/(RS)`

Thus, (3) is parallel set.