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In the diagram below, we have MN is parallel to PQ. We also know that OM=9, OQ=6, and PN=25.

Find OP

https://latex.artofproblemsolving.com/3/d/2/3d2295937acd447e223df6e9b0f7f3cf318d194f.png

 Oct 8, 2019
 #1
avatar+1076 
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If triangle OPQ's side lengths are 2/3 of OMN, then PN can be split so that OP is 2/3 of ON.

 

Basically, OP is 2/3 of ON. Together, they make up 25 (PN).

 

Can you solve from here?

 Oct 8, 2019
 #2
avatar+109450 
+2

Since MN is parallel to PQ, then triangle MNO is similar to triangle QPO

 

Then 

 

OM / OQ  =  ON/ OP

 

9/6  =  ON/ OP

 

3/2  = ON/OP

 

Then  PN  is divided into  5 equal parts  and OP  is  2 of them

 

So

 

OP  = (2/5)PN   = (2/5) 25    =   10

 

 

cool cool cool

 Oct 8, 2019

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