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ns33
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« on: January 21, 2004, 10:32:08 PM » |
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This math problem has been bugging me for almost a year and either I've been missing something for the past 12 months, or this problem is truly screwed over. -edit- This is from the American Mathematics Competitions, Contest B, issued on Feb 26, 2003. The question is #22 Let ABCD be a rhombus with AC = 16 and BD = 30. Let N be a point on AB, and let P and Q be the feet of the perpendiculars from N to AC and BD, respectively. Which of the following is closest to the minimum possible value of PQ?
(A) 6.5; (B) 6.75; © 7; (D) 7.25; (E) 7.5 This is ALL the information there is. Please do not ask for more info, and do NOT make up information without being able to show it through a standard proof. |
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« Last Edit: January 22, 2004, 09:06:11 PM by ns33 »
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evilknight
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« Reply #1 on: January 22, 2004, 02:11:37 AM » |
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the answer is 42
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 HaHa. This post cost SS .5 cents! So Sayeth the Evilknight. Hmmm. The court will take a 15 minute recess to fix the judicial...bench...thingy. ~Ethan |
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Rug
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« Reply #2 on: January 22, 2004, 10:34:21 AM » |
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Eeeek, thats evil. Maybe one of those unanswerable questions?
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opperdude
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« Reply #3 on: January 22, 2004, 10:52:50 AM » |
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well i'm just gonna start (dunno how to do the signs properly here but i'll try): to make PQ as short as possible NP=NQ (rite?) APN~NQB AN=16/46 AB=16/30 BN BN=30/46 AB=1 7/8 AN AP=16/30 NQ =16/30 NP AN=16/30 BN NP=16/30 BQ i call the point where the diagonals cross point S: AS=8 (=CS) BS=15 (=DS) ok i'll move on later, but could sum1 tell me if this is right so far? cause i'd hate it if i assumed sumthing wrong... (and if sum1 knew a way to work out sum angles it would be easy  ) |
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'Joost, like juice but better' ~fire
happy now?
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SS
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« Reply #4 on: January 22, 2004, 06:53:54 PM » |
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Can't be arsed with figuring out what opper is doing, but I'm 98.3274% certain that the solution requires the angle rules, so I'm going to go figure out if I can get anywhere using them.
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Peter 'SpectralShadows' Boughton,
Seeker of Perfection, BPsite Sitelord.
Till shade is gone, till water is gone, into the Shadow with teeth bared, screaming
defiance with the last breath, to spit in the Sightblinder's eye on the Last Day.
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RipperRoo
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« Reply #5 on: January 22, 2004, 06:58:22 PM » |
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SOHCAHTOA?
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"How could you be intimidated by a woman who had told you in dead seriousness that there were one hundred and seven different kisses, and ninety-three ways to touch a man's face with your hand?" --Min--
"Ohh my feet are getting hotter than a flame grilled otte
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SS
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« Reply #6 on: January 22, 2004, 07:01:20 PM » |
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No, that's something different and only works for right-angled triangles. I was talking about the X/Z/F thing.
Although I've just spotted a reason why it might be a really easy problem... more in a sec.
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« Last Edit: January 22, 2004, 07:03:44 PM by SS »
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Peter 'SpectralShadows' Boughton,
Seeker of Perfection, BPsite Sitelord.
Till shade is gone, till water is gone, into the Shadow with teeth bared, screaming
defiance with the last breath, to spit in the Sightblinder's eye on the Last Day.
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opperdude
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« Reply #7 on: January 22, 2004, 07:06:55 PM » |
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me's waiting with tension...
and yeah i think u need to have some trick to figure out the angles or the length of AB, and after that you'll need what i did (i think) to get to an exact number
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'Joost, like juice but better' ~fire
happy now?
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SS
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« Reply #8 on: January 22, 2004, 07:37:14 PM » |
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Gah, okay not as easy as I thought... this is tying my head in knots. :|
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Peter 'SpectralShadows' Boughton,
Seeker of Perfection, BPsite Sitelord.
Till shade is gone, till water is gone, into the Shadow with teeth bared, screaming
defiance with the last breath, to spit in the Sightblinder's eye on the Last Day.
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ns33
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« Reply #9 on: January 22, 2004, 09:05:53 PM » |
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The answer is 7.0, I figured it out yesterday. I've been assuming that its a parallelogram and not a rhombus, in which the diagonals would be perpendicular to each other. And since AB = BC = 17, it's easy to go from there. I originally thought I'd have to use law of cosines, but that wouldnt work if you arent given any other angles. I just forgot (for a year) the properties of a rhombus. :ph34r:
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SS
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« Reply #10 on: January 22, 2004, 09:11:16 PM » |
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Bah, I knew the diagonals were perpendicular, but none of the definitions mentioned it and your disclaimer made me believe you. Evil evil evil man! :angry:
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« Last Edit: January 22, 2004, 09:11:30 PM by SS »
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Peter 'SpectralShadows' Boughton,
Seeker of Perfection, BPsite Sitelord.
Till shade is gone, till water is gone, into the Shadow with teeth bared, screaming
defiance with the last breath, to spit in the Sightblinder's eye on the Last Day.
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underruler
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« Reply #11 on: January 23, 2004, 05:01:31 AM » |
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Good thing I read all your posts or I would have been plagued too...I figured out that the perpendiculars were diagonal, but it's been quite some time since I've had some geometry problems. I'm so rusty. *creak creak*
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