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The vertices of a 3-4-5 right triangle are the centers of three mutual

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The vertices of a 3-4-5 right triangle are the centers of three mutual  [#permalink]

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New post 25 Mar 2019, 01:16
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A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

58% (01:39) correct 42% (02:42) wrong based on 19 sessions

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The vertices of a 3-4-5 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?


(A) \(12\pi\)

(B) \(\frac{25\pi}{2}\)

(C) \(13\pi\)

(D) \(\frac{27\pi}{2}\)

(E) \(14\pi\)

Attachment:
2006_AMC_12A_Problem_13.gif
2006_AMC_12A_Problem_13.gif [ 9.14 KiB | Viewed 319 times ]

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Re: The vertices of a 3-4-5 right triangle are the centers of three mutual  [#permalink]

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New post 25 Mar 2019, 02:01
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Bunuel wrote:
Image
The vertices of a 3-4-5 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?


(A) \(12\pi\)

(B) \(\frac{25\pi}{2}\)

(C) \(13\pi\)

(D) \(\frac{27\pi}{2}\)

(E) \(14\pi\)

Attachment:
2006_AMC_12A_Problem_13.gif


Let us consider the radius of the circles to be:
Smallest circle = x (bottom left)
Medium = y (top)
Biggest = z (bottom right)

Since this is a 3-4-5 circle, then we can deduce the sides to be:

x + y = 3
x + z = 4
y + z = 5

Adding all three: 2(x + y + z) = 3 + 4 + 5 = 12
=> x + y + z = 6
=> x = 1, y = 2, z = 3
Area of 3 circles = pie (x^2 + y^2 + z^2) = pie (3^3 + 2^2 + 1^2) = 14pie

Hence E is the correct answer.
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Re: The vertices of a 3-4-5 right triangle are the centers of three mutual  [#permalink]

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New post 25 Mar 2019, 03:04
Bunuel wrote:
Image
The vertices of a 3-4-5 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?


(A) \(12\pi\)

(B) \(\frac{25\pi}{2}\)

(C) \(13\pi\)

(D) \(\frac{27\pi}{2}\)

(E) \(14\pi\)

Attachment:
2006_AMC_12A_Problem_13.gif


we can solve <30 sec
visualize the given image
the largest length = 5
smallest = 3
and third side = 4
the value of radius can be 3,2,1 ; square = 9pi+4pi+1pi = 14 pi
IMO E
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Re: The vertices of a 3-4-5 right triangle are the centers of three mutual   [#permalink] 25 Mar 2019, 03:04
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