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In the figure above, line segment AD is the diameter of circle O, line

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In the figure above, line segment AD is the diameter of circle O, line [#permalink]

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New post 09 Aug 2017, 23:36
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In the figure above, line segment AD is the diameter of circle O, line segment AO is the diameter of circle B, line segment OD is the diameter of circle C, and circle E is tangent to each of the other circles. If the radius of circle O is 4, what is the radius of circle E?

A. 2/3

B. 3/4

C. 1

D. 4/3

E. 3/2

[Reveal] Spoiler:
Attachment:
FourCircles.png
FourCircles.png [ 11.47 KiB | Viewed 927 times ]
[Reveal] Spoiler: OA

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In the figure above, line segment AD is the diameter of circle O, line [#permalink]

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New post 10 Aug 2017, 00:09
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Radius OA=4
therefore OB=OC=2

Now let radius of circle E =x

BE=EC=2+x

As BE=EC and OB=OC, implies EO is perpendicular to BC

Now in triangle EOB :
OB=2
EB=2+x
EO =4-x (radius of circle O - radius of circle E)

solving Pythagoras : OB^2+OE^2=BE^2
=> x=4/3

Answer : D

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Re: In the figure above, line segment AD is the diameter of circle O, line [#permalink]

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New post 15 Aug 2017, 10:58
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Bunuel wrote:
Image
In the figure above, line segment AD is the diameter of circle O, line segment AO is the diameter of circle B, line segment OD is the diameter of circle C, and circle E is tangent to each of the other circles. If the radius of circle O is 4, what is the radius of circle E?

A. 2/3

B. 3/4

C. 1

D. 4/3

E. 3/2

[Reveal] Spoiler:
Attachment:
FourCircles.png


Since the radius of circle O is 4 (i.e., AO = 4), BO = 2. We can form a right triangle by connecting B, E, and O. If we let x = radius of circle E, the legs of right triangle BOE are BO = 2 and OE = 4 - x, and its hypotenuse is BE = 2 + x. Now, using the Pythagorean theorem, we have:

2^2 + (4 - x)^2 = (2 + x)^2

4 + 16 - 8x + x^2 = 4 + 4x + x^2

16 - 8x = 4x

16 = 12x

x = 16/12 = 4/3

Answer: D
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Kudos [?]: 1010 [1], given: 3

Re: In the figure above, line segment AD is the diameter of circle O, line   [#permalink] 15 Aug 2017, 10:58
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In the figure above, line segment AD is the diameter of circle O, line

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