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In the diagram, AB = 8 and AD = 9. Two circles are tangent to each oth

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In the diagram, AB = 8 and AD = 9. Two circles are tangent to each oth  [#permalink]

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New post 25 Jul 2018, 00:35
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A
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C
D
E

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23% (03:46) correct 77% (02:43) wrong based on 63 sessions

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In the diagram, AB = 8 and AD = 9. Two circles are tangent to each other and also tangent to the sides of the rectangle, as shown. If the radius of the smaller circle is 2, what is the radius of the larger circle?


(A) \(\frac{5}{2}\)

(B) \(2\sqrt{2}\)

(C) \(3\)

(D) \(2\sqrt{3}\)

(E) \(\frac{7}{2}\)


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Re: In the diagram, AB = 8 and AD = 9. Two circles are tangent to each oth  [#permalink]

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New post 25 Jul 2018, 03:26
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3
OA: C
Attachment:
circle.PNG
circle.PNG [ 39.79 KiB | Viewed 844 times ]

Let the radius of big circle be R , Then \(AI =BG=R\)
Small circle radius \(=O_{S}E=O_{S}F=JG=HB=2cm\)
\(AB=8=AI+IH+HB=R+IH+2\)
\(IH=8-2-R =6-R\)

\(AD=BC= 9\)
\(BC=BG+GF+FC=R+GF+2\)
\(GF =9-2-R =7-R\)

\(O_{S}O_{B}=R+2\)

In \(\triangle O_{S}O_{B}J\),

\((O_{S}O_{B})^2 = (O_{S}J)^2 +(O_{B}J)^2\)
\(O_{S}J = GF\)
\(O_{B}J = IH\)

\((R+2)^2=(6-R)^2+(7-R)^2\)
\(R^2+4+4R = 36 +R^2 -12R +49 +R^2 -14R\)
\(R^2+4R+4 = 2R^2 -26R +85\)
\(R^2 -30R +81=0\)
Solving for R , we get R=3cm or 27 cm( neglecting 27 cm as it is not possible to have circle with R=27 inscribed inside 8*9rectangle)
Final \(R=3 cm\)
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Re: In the diagram, AB = 8 and AD = 9. Two circles are tangent to each oth  [#permalink]

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New post 06 Oct 2018, 07:47
Princ wrote:
OA: C
Attachment:
circle.PNG

Let the radius of big circle be R , Then \(AI =BG=R\)
Small circle radius \(=O_{S}E=O_{S}F=JG=HB=2cm\)
\(AB=8=AI+IH+HB=R+IH+2\)
\(IH=8-2-R =6-R\)

\(AD=BC= 9\)
\(BC=BG+GF+FC=R+GF+2\)
\(GF =9-2-R =7-R\)

\(O_{S}O_{B}=R+2\)

In \(\triangle O_{S}O_{B}J\),

\((O_{S}O_{B})^2 = (O_{S}J)^2 +(O_{B}J)^2\)
\(O_{S}J = GF\)
\(O_{B}J = IH\)

\((R+2)^2=(6-R)^2+(7-R)^2\)
\(R^2+4+4R = 36 +R^2 -12R +49 +R^2 -14R\)
\(R^2+4R+4 = 2R^2 -26R +85\)
\(R^2 -30R +81=0\)
Solving for R , we get R=3cm or 27 cm( neglecting 27 cm as it is not possible to have circle with R=27 inscribed inside 8*9rectangle)
Final \(R=3 cm\)



@bunnel Is this explanation even correct i mean is this method correct ?
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In the diagram, AB = 8 and AD = 9. Two circles are tangent to each oth  [#permalink]

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New post 06 Oct 2018, 12:08
1
Diagonal of the rectangle = √(\(8^2 + 9^2\)) = √145
Let the radius of larger circle be x.

x√2 + x + 2√2 + 2 = √145 (diagonal of larger square + radius of larger circle + diagonal of smaller square + radius of smaller circle)

2.4 x = 12.1-4.82
x = 7.2/2.414 = 2..98.. = 3

Ans C
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In the diagram, AB = 8 and AD = 9. Two circles are tangent to each oth   [#permalink] 06 Oct 2018, 12:08
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