+1836 \(In the diagram below, we have $\overline{MN}\parallel\overline{PQ}$, $OM = 9$, $OQ = 6$, and $PN = 24$. Find $OP$. [asy] pair O,M,NN,P,Q; M = (-0.4,0.5); NN = (0.3,0.5); P = -0.9*NN; Q = -0.9*M; label("$O$",O,E); label("$M$", M,N); label("$N$", NN,N); label("$P$",P,S); label("$Q$",Q,S); draw(M--NN--P--Q--M); [/asy]\)In the diagram below, we have , , , and . Find .
+1036 LancelotLink turns into a really POd chimp when I hire away his de/coders–especially if I don’t cross his palm with a few bananas. He’s almost as POed when he catches Sir LaTex gold-bricking –he’s very skilled at it. Lancelot throws banana peels at him after catching him perusing his stamp collection on company time.
Along with roasted peanuts, Sir La-Tex likes the occasional mouse, but I am uncertain if he likes shrimp/prawns. I’ve not had BBQed prawns before, but I often have BBQed crawdads –or eat them raw – they are a troll staple and easily found near troll toll bridges. I may try some BBQed prawns the next time I BBQ a steak for my dogs. We dogs (and trolls) really like our steaks.
+1036 The embedded (La)tex compiler for this forum does not render Asy(mptote) code.
In the diagram below, we have $\overline{MN}\parallel\overline{PQ}$, $OM = 9$, $OQ = 6$, and $PN = 24$. Find $OP$. [asy] pair O,M,NN,P,Q; M = (-0.4,0.5); NN = (0.3,0.5); P = -0.9*NN; Q = -0.9*M; label("$O$",O,E); label("$M$", M,N); label("$N$", NN,N); label("$P$",P,S); label("$Q$",Q,S); draw(M--NN--P--Q--M); [/asy]
Here’s image of the rendered code:
+118673 Thanks Nauseated.
MN is parallel to PQ
Consider triangle MNO and triangle QPO
< MOP = < QOM Vertiacally opposite angles are congruent.
<MNO = < QPO Alternate angles on parallel lines are congruent
<NMO = < PQO Alternate angles on parallel lines are congruent
Therefore triangle MNO and triangle QPO (3 equal angles.)
Therefore
\(\frac{0M}{ON}=\frac{OQ}{OP}\\ \frac{9}{ON}=\frac{6}{OP}\\ \frac{9}{6}=\frac{ON}{OP}\\ \frac{3}{2}=\frac{ON}{OP}\\ But \;NP=24\\ \mbox{We need to divide 24 in the ratio 3:2}\\ 3/5*24 = 14.4\\ 2/5*24 = 9.6\\ So\\ ON=14.4\;units\;\;\; and\;\;\; OP=9.6\;units\)
Mellie, if you do not understand you need to ask questions :)
+129852 Triangle MON ≈ Triangle QOP by AA congruency....therefore.....
OM / ON = OQ / OP Let ON = x.....then OP = 24 - x.......and we have
9 / x = 6 / [ 24 - x] cross-multiply
9 [ 24 - x ] = 6x simplify
216 - 9x = 6x add 9x to both sides
216 = 15 x
x = 216/15 = 14.4 = ON
And OP = 24 - 14.4 = 9.6
+129852 Don't worry, Nauseated.....I think I have found the perfect "LaTex" detective amongst your brood of monkeys.....Sir "LA-TeX"......that name is not a coincidence......he actually has legal residences in both states.......he's a whiz at deciphering the indecipherable.......he's working on Mellie's codes as we speak........having some experience with cracking the Japanese "Purple Codes" and the German "Enigma Codes" in WWII.....this should be cinch for him.....!!!
+118673 WOW Chris, this is great. We now have Sir La-TeX working with us.
I know he works for bananas too!!
Just as well we have no greenbacks or Aussie plastic to pay him with!
We could probably throw in a few peanuts though. I know chimps are partial to peanuts.
Welcome to the team Sir La-TeX
+118673
Australians do not have shrimps. We have prawns and we rarely BBQ them.
You have been watching too much of Paul Hogan's silly ads!! Shame on you!!
Anyway, as much as La-TeX may like prawns, they are not good for him.
If we allow him to eat rubbish he will get sick and then we will not get our peanuts and bananas worth out of him.
NO La-TeX will not be fed prawns on my watch!
+1036 LancelotLink turns into a really POd chimp when I hire away his de/coders–especially if I don’t cross his palm with a few bananas. He’s almost as POed when he catches Sir LaTex gold-bricking –he’s very skilled at it. Lancelot throws banana peels at him after catching him perusing his stamp collection on company time.
Along with roasted peanuts, Sir La-Tex likes the occasional mouse, but I am uncertain if he likes shrimp/prawns. I’ve not had BBQed prawns before, but I often have BBQed crawdads –or eat them raw – they are a troll staple and easily found near troll toll bridges. I may try some BBQed prawns the next time I BBQ a steak for my dogs. We dogs (and trolls) really like our steaks.
+118673 I knew Sir La-TeX is a very generous philanthropist, He has agreeed to help us I believe. If he is a gold bricker sometimes that is ok, Sir Lancelot should not come down on him to hard. He is working for bananas after all! I suppose when a chimp has the name like Sir Phil LaTeX it is very obvious that he would be a philatelist! I have some old stamps he may be interested in seeing! Give him a little space. Let him study his stamps. He may find some great code in there somewhere.
I think any Troll who is worth his salt would like sushi. He probably would like it best when it has ben naturally putrified by the resident croc or gator. Crocs and gators like to stow there dinner in underwater crevices and come back for it in a few weeks when it is soft and ripe for they have no teeth to chew it with. I think any self respecting troll would like it that way too! Steak can be prepared in this way too 
