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The above diagram shows 3 identical circles placed inside a right

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by Brent@GMATPrepNow » Wed Mar 11, 2020 3:12 pm

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Difficulty—

Image

The above diagram shows 3 identical circles placed inside a right triangle so that all 3 circles are tangent to line AB. What is the radius of each circle?

A) 5
B) 6
C) 7
D) 8
E) 10

Answer: A
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Re: The above diagram shows 3 identical circles placed inside a right

by Brent@GMATPrepNow » Thu Mar 12, 2020 5:10 am
Brent@GMATPrepNow wrote: ↑
Wed Mar 11, 2020 3:12 pm
 Image

The above diagram shows 3 identical circles placed inside a right triangle so that all 3 circles are tangent to line AB. What is the radius of each circle?

A) 5
B) 6
C) 7
D) 8
E) 10

Answer: A
Source: www.gmatprepnow.com
Key property: A line tangent to a circle is perpendicular to the circle's radius at the point of intersection

First, if we add the centers of each triangle as as well as the point E, we get the following:
Image


At this point, if we draw lines from point E to each of the three vertices, look at the following:
Image


At this point we may recognize that: (area of ∆AEB) + (area of ∆CEA) + (area of ∆CEB) = area of ∆ACB

Area of triangle = (base)(height)/2

So, we get: (40)(r)/2 + (30)(5r)/2 + (50)(r)/2 = (40)(30)/2
Simplify to get: 20r + 75r + 25r = 600
Simplify: 120r = 600
Solve: r = 600/120 = 5

Answer: A

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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