Geometry question? (1 Viewer)

[rsingh]

Hey guys, I have a tough geometry question ...

The side LM of a triangle LMN is produced to P so that MP=0.5LM. If Q and R are the midpoints of MN and LN respectively, prove that PQ is parallel to MR.

Any help would be greatly appreciated.
Thanks!

 

[shafqat]

I don't know if this is the best method, but:

Let S be the midpt of LM. Construct RS.

Prove SR is parallel to MN (easily done considering triangles LSR and LMN are similar)

Now we know angle RSM = QMP

Also RS/SM = QM/MP (prove before hand that RS = 1/2MN from the same similar triangles as above)

triangle RSM similar to QMP

so angle SRM = MQP

but angle SRM = RMQ (SR is parallel to MN)

so angle RMQ = MQP

so RM is parallel tp PQ (alternate angles equal)