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

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by BTGmoderatorDC » Mon Apr 22, 2019 6:27 pm

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A

B

C

D

E

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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

OA D

Source: Veritas Prep
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by Jay@ManhattanReview » Mon Apr 22, 2019 10:13 pm
BTGmoderatorDC 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

OA D

Source: Veritas Prep
We have

OA = Radius of circle O = 4; thus, OB = OC = 2

Say the radius of circle E = x

Thus, BE =2 + x

In the ∆EOB, EO is perpendicular to OB, thus, /_EOB = 90º and EO = Radius of circle O - x = 4 - x

Thus, ∆EOB is a right angle triangle. Applying Pythagoras theorem, we have

BE^2 + EO^2 + OB^2

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

=> x = 4/3

The correct answer: D

Hope this helps!

-Jay
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