Problem Solving

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]

@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..

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

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

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).

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 wrote:

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

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).

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]

finally an error in summation...this is miserable ..still correcting the original post

sanju09 wrote:

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

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).

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]

Good Quotation of Edward Richards ; I hope you follow those words in letter and spirit !!!!

Good Quotation of Edward Richards ; I hope you follow those words in letter and spirit !!!!

I only wish to be WISE, neither OLD nor OWL

sanju09 wrote:

Good Quotation of Edward Richards ; I hope you follow those words in letter and spirit !!!!

I only wish to be WISE, neither OLD nor OWL

" THE LESS HE SPOKE , THE MORE HE HEARD " That's the fundamental message , by Edward Richards !!!

Haldiram Bhujiawala wrote:

sanju09 wrote:

Good Quotation of Edward Richards ; I hope you follow those words in letter and spirit !!!!

I only wish to be WISE, neither OLD nor OWL

" THE LESS HE SPOKE , THE MORE HE HEARD " That's the fundamental message , by Edward Richards !!!

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?

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!!

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?

Some weird interpretaion indeed !!!

gmatmachoman 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!!

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 ji

Haldiram Bhujiawala wrote:

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?

Some weird interpretaion indeed !!!

sorry if it didn't hurt an owl