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Math Expert V
Joined: 02 Sep 2009
Posts: 58402
The vertices of a 3-4-5 right triangle are the centers of three mutual  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 67% (01:50) correct 33% (02:42) wrong based on 24 sessions

HideShow timer Statistics 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 [ 9.14 KiB | Viewed 422 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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Bunuel wrote: 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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Bunuel wrote: 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 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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