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# Circles A, B and C are externally tangent to each other, and internall

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Math Expert
Joined: 02 Sep 2009
Posts: 58396
Circles A, B and C are externally tangent to each other, and internall  [#permalink]

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21 Mar 2019, 05:05
00:00

Difficulty:

95% (hard)

Question Stats:

18% (04:44) correct 82% (02:29) wrong based on 11 sessions

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Circles A, B and C are externally tangent to each other, and internally tangent to circle D. Circles B and C are congruent. Circle A has radius 1 and passes through the center of D. What is the radius of circle B?

(A) 2/3

(B) $$\frac{\sqrt{3}}{2}$$

(C) 7/8

(D) 8/9

(E) $$\frac{1 + \sqrt{3}}{3}$$

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db04d82b5f5ca4618bfc01360dadf7cd5388f624.png [ 25.72 KiB | Viewed 685 times ]

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Math Expert
Joined: 02 Aug 2009
Posts: 8004
Circles A, B and C are externally tangent to each other, and internall  [#permalink]

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26 Mar 2019, 09:03
[quote="Bunuel"]Circles A, B and C are externally tangent to each other, and internally tangent to circle D. Circles B and C are congruent. Circle A has radius 1 and passes through the center of D. What is the radius of circle B?

(A) 2/3

(B) $$\frac{\sqrt{3}}{2}$$

(C) 7/8

(D) 8/9

(E) $$\frac{1 + \sqrt{3}}{3}$$

Join the lines as shown...
Now DE = 2, double the radius of circle A. So, BD = DE-BE=2-r
Take triangle BCD, $$CD=\sqrt{(2-r)^2-r^2}=\sqrt{4+r^2-4r-r^2}=2\sqrt{1-r}$$..
Now, take triangle ABC, $$AB^2=(BC)^2+(AD+DC)^2..........(1+r)^2=r^2+(1+2\sqrt{1-r})^2....1+2r+r^2=r^2+1+4(1-r)+4\sqrt{1-r}....9r^2=8r$$. r cannot be 0, so 9r=8 or r=$$\frac{8}{9}$$

D
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db04d82b5f5ca4618bfc01360dadf7cd5388f624.png [ 29.36 KiB | Viewed 394 times ]

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Circles A, B and C are externally tangent to each other, and internall   [#permalink] 26 Mar 2019, 09:03
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