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# Points P, Q, R, S, and T all lie on the same line. The larger circle

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Points P, Q, R, S, and T all lie on the same line. The larger circle  [#permalink]

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11 Mar 2015, 04:48
2
3
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55% (hard)

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65% (02:27) correct 35% (02:30) wrong based on 133 sessions

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Points P, Q, R, S, and T all lie on the same line. The larger circle has center S and passes through P and T. The smaller circle has center R and passes through Q and S. What is the ratio of the area of the larger circle to the area of the smaller circle?

(1) ST:PQ = 5/2
(2) RT:PR = 13/7

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Re: Points P, Q, R, S, and T all lie on the same line. The larger circle  [#permalink]

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11 Mar 2015, 07:08
1

from statment 1 we have that ST:PQ = 5x/2x
PT is the diameter of large circle and ST=PS =Radius of large circle

PQ+QR+RS=ST
2x+QR+RS=5x
QR+RS=3x we know that QR=RS=Radius of small circle
QR=RS=3x/2

so the area of large circle = pir^2=pi(5x)= pi 25x^2

the area of small circle=pi(3x/2)^2=9x^2/4

so the ratio of the area of the larger circle to the area of the smaller circle=25x^2*4/9x^2=100/9.

statment 2:we have that RT:PR = 13x/7x
RT+PR=20x=diameter of large circle
PS=ST=10x
10x-7x=3x
so the ratio of the area of the larger circle to the area of the smaller circle pi(10x)^2/pi(3x)^2=100/9
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Re: Points P, Q, R, S, and T all lie on the same line. The larger circle  [#permalink]

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11 Mar 2015, 07:13
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To find the ratio of area of two circles, we need ratio of their radii.
Statement 1 and 2 both are Sufficient to get the ratio of the radii.
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Re: Points P, Q, R, S, and T all lie on the same line. The larger circle  [#permalink]

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11 Mar 2015, 09:38
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Hi

The pic attached explains
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n1.png [ 12.04 KiB | Viewed 3278 times ]

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Re: Points P, Q, R, S, and T all lie on the same line. The larger circle  [#permalink]

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15 Mar 2015, 22:49
Bunuel wrote:

Points P, Q, R, S, and T all lie on the same line. The larger circle has center S and passes through P and T. The smaller circle has center R and passes through Q and S. What is the ratio of the area of the larger circle to the area of the smaller circle?

(1) ST:PQ = 5/2
(2) RT:PR = 13/7

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

Call the radius of the larger circle y, and y = PS = ST. Call the radius of the smaller circle x, and x = RS = RQ. If we took a ratio of the areas, the factors of pi would cancel and we would be left with the ratio (y/x) squared. If we could solve for this simpler ratio, y/x, then we could find the ratio of areas.

Statement #1: ST = y and PQ = y – 2x, so
Attachment:

cpotg_img17.png [ 5.19 KiB | Viewed 3159 times ]

This allows us to solve for the ratio y/x, which would allow us to find the ratio of areas. This statement, alone and by itself, is sufficient.

Statement #2: RT = y + x and PR = y – x, so
Attachment:

cpotg_img18.png [ 1.64 KiB | Viewed 3156 times ]

Cross-multiply.
Attachment:

cpotg_img19.png [ 2.75 KiB | Viewed 3157 times ]

This allows us to solve for the ratio y/x, which would allow us to find the ratio of areas. This statement, alone and by itself, is sufficient.

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Re: Points P, Q, R, S, and T all lie on the same line. The larger circle  [#permalink]

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25 Jun 2019, 14:44
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Re: Points P, Q, R, S, and T all lie on the same line. The larger circle   [#permalink] 25 Jun 2019, 14:44
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