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# In circle P, the two chords intersect at point X, with the lengths as

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Math Expert
Joined: 02 Sep 2009
Posts: 51098
In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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16 Feb 2016, 00:07
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Difficulty:

95% (hard)

Question Stats:

46% (02:24) correct 54% (02:19) wrong based on 90 sessions

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In circle P, the two chords intersect at point X, with the lengths as indicated in the figure. Which could not be the sum of lengths a and b, if a and b are integers?

A. 49
B. 30
C. 26
D. 16
E. 14

Attachment:

2016-02-14_1430.png [ 8.49 KiB | Viewed 1882 times ]

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Math Expert
Joined: 02 Aug 2009
Posts: 7100
In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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16 Feb 2016, 01:01
2
1
Bunuel wrote:

In circle P, the two chords intersect at point X, with the lengths as indicated in the figure. Which could not be the sum of lengths a and b, if a and b are integers?

A. 49
B. 30
C. 26
D. 16
E. 14

Attachment:
The attachment 2016-02-14_1430.png is no longer available

Hi,
To do this the CHORD INTERSECTING THEOREM is the simplest..
Quote:
Given a point X in the interior of a circle, pass two lines through X that intersect the circle in points A and B and, respectively, D and C. Then AX·BX = DX·CX

How do you get it or if you do not know the theorem, how do you get to the answer..

Another property of CHORDS
Quote:
A chord will subtend the same angle from any point on the circumference on the same side of chord

so see the att sketch..

ACX and DBX are similar triangles and we get AX*BX=DX*CX..
substitute values of AX and BX...
6*8= DX*CX..
so a*b= 48...
we have to find the sum of a and b, so lets test the values..

A. 49..ab=48 = 1*48.. so POSSIBLE if a and b are 1 and 48, a+b=49
B. 30..ab=30... no values of a and b possible as integer
C. 26..ab=26 = 2*24.. so POSSIBLE if a and b are 2 and 24, a+b=26
D. 16..ab=16 = 4*12.. so POSSIBLE if a and b are 4 and 12, a+b=16
E. 14..ab=14 = 6*8.. so POSSIBLE if a and b are 6 and 8, a+b=14
ans B..

also we can check values by factoring..
48=1*2*2*2*2*3...
largest sum possible if it is product of 1 and the integer itself, 48*1, so sum = 1+48=49..
next largest sum is when it is product of 2nd smallest factor,2=24*2.. and sum =2+24=26..

here we can see 30 is larger than 26, so not possible..

Attachments

IMG_5546.JPG [ 2.04 MiB | Viewed 2071 times ]

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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Joined: 22 Sep 2015
Posts: 97
Re: In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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22 Feb 2016, 15:27
E: 14 because the sum of the shorter cord is 14, so the longer chord cannot be that.
Intern
Joined: 08 Feb 2015
Posts: 23
In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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29 Feb 2016, 07:16
Bunuel wrote:

In circle P, the two chords intersect at point X, with the lengths as indicated in the figure. Which could not be the sum of lengths a and b, if a and b are integers?

A. 49
B. 30
C. 26
D. 16
E. 14

Attachment:
2016-02-14_1430.png

Hi Bunuel,

Should D be 19?

We know that the triangle is having a defined ratio

6 -> b = 3 or 2 or 1 (divide by 2, 3 and 6 respectively since b must be int)
8 -> a = 16 or 24 or 48 ( times by 2, 3 and 6 following the above)

hence possible values 14, 19, 26 or 49... I don't see how we can get a+b=16 and the other possible answer is 30...
Manager
Joined: 16 Feb 2016
Posts: 52
Concentration: Other, Other
Re: In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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29 Feb 2016, 07:23
Answered this by substitution in 20 seconds.

6x8=48
So now get the combinations for 48:

1 and 48; sum is 49
2 and 24; sum is 26
3 and 16; sum is 19
4 and 12; sum is 16
6 and 8 ;sum is 14

The only one missing is 30 > Answer B
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Posts: 9106
Re: In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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25 Jun 2018, 04:59
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Re: In circle P, the two chords intersect at point X, with the lengths as &nbs [#permalink] 25 Jun 2018, 04:59
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