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

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

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New post 16 Feb 2016, 01:07
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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
2016-02-14_1430.png [ 8.49 KiB | Viewed 1816 times ]

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

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New post 16 Feb 2016, 02:01
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Bunuel wrote:
Image
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..


Image
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..

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Re: In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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New post 22 Feb 2016, 16:27
E: 14 because the sum of the shorter cord is 14, so the longer chord cannot be that.
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In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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New post 29 Feb 2016, 08:16
Bunuel wrote:
Image
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...
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Re: In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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New post 29 Feb 2016, 08:23
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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Re: In circle P, the two chords intersect at point X, with the lengths as  [#permalink]

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New post 25 Jun 2018, 05:59
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Re: In circle P, the two chords intersect at point X, with the lengths as &nbs [#permalink] 25 Jun 2018, 05:59
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