The points A, B, C lie on a line in that order with AB = 9, BC = 21. Let O be a point not on AC such that AO = CO and the distances AO and BO are integral. What is the sum of all possible perimeters of triangle ACO?
(A) 320
(B) 350
(C) 380
(D) 410
(E) 420
[spoiler]My answer: 128, choices...? source: BTG PS Forum[/spoiler]
point not on AC
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- sanju09
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Last edited by sanju09 on Wed May 19, 2010 6:28 am, edited 1 time in total.
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@sanju09..great question..as of now I am able to deduce only this
the point O has to be on perpendicular bisector line of AC at distance of 15
let it intersect AC at P
so AP=15 and BP=6
6 can form only one right angle triangle when its all three sides are integer 6,8,10
so point on the bisector is 8 unit away from P and 17 unit away from A or C...perimeter= 2*(17+17+30) or 128
but this is not in ans..am I in right track or..its time for a break..
the point O has to be on perpendicular bisector line of AC at distance of 15
let it intersect AC at P
so AP=15 and BP=6
6 can form only one right angle triangle when its all three sides are integer 6,8,10
so point on the bisector is 8 unit away from P and 17 unit away from A or C...perimeter= 2*(17+17+30) or 128
but this is not in ans..am I in right track or..its time for a break..
"If you don't know where you are going, any road will get you there."
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ok..here is the complete solution,
let distance from point be x and that from point B be y...
so x^2-15^2=y^2-6^2
or (x+y)(x-y)=189
now 189 can be expressed as multiple of 2 integers in the following ways (1,189),(3,63),(7,27),(9,21) since x>y we get
(x+y)----(x-y)----2x
189-------1-------190
63---------3-------66
27---------7-------34
21---------9-------30......this will be the case when the point in on AC
So sum of all possible perimeters is 190+66+34+3*30=380
ans option C
let distance from point be x and that from point B be y...
so x^2-15^2=y^2-6^2
or (x+y)(x-y)=189
now 189 can be expressed as multiple of 2 integers in the following ways (1,189),(3,63),(7,27),(9,21) since x>y we get
(x+y)----(x-y)----2x
189-------1-------190
63---------3-------66
27---------7-------34
21---------9-------30......this will be the case when the point in on AC
So sum of all possible perimeters is 190+66+34+3*30=380
ans option C
Last edited by liferocks on Thu May 20, 2010 12:49 am, edited 1 time in total.
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I must say that you are a real fighter, and hence you finally proved one of the choices as correct. Good part about your work is that all possibilities shown by you maintain the most important feature of triangle ACO, which is, AO > 15. Everything is perfect EXCEPT the [spoiler]answer, which in my humble opinion, should be (C) instead of (A).liferocks wrote:ok..here is the complete solution,
let distance from point be x and that from point B be y...
so x^2-15^2=y^2-6^2
or (x+y)(x-y)=189
now 189 can be expressed as multiple of 2 integers in the following ways (1,189),(3,63),(7,27),(9,21) since x>y we get
(x+y)----(x-y)----2x
189-------1-------190
63---------3-------66
27---------7-------34
21---------9-------30......this will be the case when the point in on AC
So sum of all possible perimeters is 190+66+34+3*30=320
ans option A
Nonetheless, you have been included to the hit list of my Quant Commandos on BTG, which reads
Many old members followed by...gmatmachoman, shashank.ism, thephoenix, ajith, harshavardhanc, rockeyb,...and with few present names definitely missing, and finally it's you liferocks. You certainly deserve my thanks too![/spoiler]
The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
Sanjeev K Saxena
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finally an error in summation...this is miserable ..still correcting the original postsanju09 wrote:I must say that you are a real fighter, and hence you finally proved one of the choices as correct. Good part about your work is that all possibilities shown by you maintain the most important feature of triangle ACO, which is, AO > 15. Everything is perfect EXCEPT the [spoiler]answer, which in my humble opinion, should be (C) instead of (A).liferocks wrote:ok..here is the complete solution,
let distance from point be x and that from point B be y...
so x^2-15^2=y^2-6^2
or (x+y)(x-y)=189
now 189 can be expressed as multiple of 2 integers in the following ways (1,189),(3,63),(7,27),(9,21) since x>y we get
(x+y)----(x-y)----2x
189-------1-------190
63---------3-------66
27---------7-------34
21---------9-------30......this will be the case when the point in on AC
So sum of all possible perimeters is 190+66+34+3*30=320
ans option A
Nonetheless, you have been included to the hit list of my Quant Commandos on BTG, which reads
Many old members followed by...gmatmachoman, shashank.ism, thephoenix, ajith, harshavardhanc, rockeyb,...and with few present names definitely missing, and finally it's you liferocks. You certainly deserve my thanks too![/spoiler]
"If you don't know where you are going, any road will get you there."
Lewis Carroll
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Good Quotation of Edward Richards ; I hope you follow those words in letter and spirit !!!!sanju09 wrote:I must say that you are a real fighter, and hence you finally proved one of the choices as correct. Good part about your work is that all possibilities shown by you maintain the most important feature of triangle ACO, which is, AO > 15. Everything is perfect EXCEPT the [spoiler]answer, which in my humble opinion, should be (C) instead of (A).liferocks wrote:ok..here is the complete solution,
let distance from point be x and that from point B be y...
so x^2-15^2=y^2-6^2
or (x+y)(x-y)=189
now 189 can be expressed as multiple of 2 integers in the following ways (1,189),(3,63),(7,27),(9,21) since x>y we get
(x+y)----(x-y)----2x
189-------1-------190
63---------3-------66
27---------7-------34
21---------9-------30......this will be the case when the point in on AC
So sum of all possible perimeters is 190+66+34+3*30=320
ans option A
Nonetheless, you have been included to the hit list of my Quant Commandos on BTG, which reads
Many old members followed by...gmatmachoman, shashank.ism, thephoenix, ajith, harshavardhanc, rockeyb,...and with few present names definitely missing, and finally it's you liferocks. You certainly deserve my thanks too![/spoiler]
- sanju09
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I only wish to be WISE, neither OLD nor OWLGood Quotation of Edward Richards ; I hope you follow those words in letter and spirit !!!!
The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
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" THE LESS HE SPOKE , THE MORE HE HEARD " That's the fundamental message , by Edward Richards !!!sanju09 wrote:I only wish to be WISE, neither OLD nor OWLGood Quotation of Edward Richards ; I hope you follow those words in letter and spirit !!!!
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No one likes to hear an owl speaking, that's why he's got to speak less, and the curse turns to boon when he gets a fairer chance to hear (learn) than the others. Isn't that the fundamental message, by Edward H. Richards?Haldiram Bhujiawala wrote:" THE LESS HE SPOKE , THE MORE HE HEARD " That's the fundamental message , by Edward Richards !!!sanju09 wrote:I only wish to be WISE, neither OLD nor OWLGood Quotation of Edward Richards ; I hope you follow those words in letter and spirit !!!!
The mind is everything. What you think you become. -Lord Buddha
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Sanju bhai!
Thx for the kind words...getting a "comment" that too positive one is welcome....
U forgot to mention fibbonacci .......She is one real fighter like Liferocks!!
I wish we all should finish off GMAT in a smoother succesful fashion!!
Thx for the kind words...getting a "comment" that too positive one is welcome....
U forgot to mention fibbonacci .......She is one real fighter like Liferocks!!
I wish we all should finish off GMAT in a smoother succesful fashion!!
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Some weird interpretaion indeed !!!sanju09 wrote:
No one likes to hear an owl speaking, that's why he's got to speak less, and the curse turns to boon when he gets a fairer chance to hear (learn) than the others. Isn't that the fundamental message, by Edward H. Richards?
- sanju09
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please wait till I release the original list of my Quant Commandos on BTG, then tell who really went unnoticed from my inner sight, Govindacharya jigmatmachoman wrote:Sanju bhai!
Thx for the kind words...getting a "comment" that too positive one is welcome....
U forgot to mention fibbonacci .......She is one real fighter like Liferocks!!
I wish we all should finish off GMAT in a smoother succesful fashion!!
The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
- sanju09
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sorry if it didn't hurt an owlHaldiram Bhujiawala wrote:Some weird interpretaion indeed !!!sanju09 wrote:
No one likes to hear an owl speaking, that's why he's got to speak less, and the curse turns to boon when he gets a fairer chance to hear (learn) than the others. Isn't that the fundamental message, by Edward H. Richards?
The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com