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In the following figure, AB is the diameter of semicircle with center

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In the following figure, AB is the diameter of semicircle with center  [#permalink]

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New post 12 Jul 2018, 21:10
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In the following figure, AB is the diameter of semicircle with center M. Two semi-circles are drawn with AM and MB as the diameters. A circle is drawn such that it is tangent to all the three semi-circles. If AM = \(r\) units, find the radius of the circle (in terms of \(r\)).
Image


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

B) \(\frac{r}{3}\)

C) \(\frac{r}{4}\)

D) \(\frac{r}{{\sqrt{2}}}\)

E) \(\frac{2r}{5}\)


Attachment:
semi.PNG
semi.PNG [ 16.86 KiB | Viewed 371 times ]

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In the following figure, AB is the diameter of semicircle with center  [#permalink]

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New post Updated on: 12 Jul 2018, 22:36
Princ wrote:
In the following figure, AB is the diameter of semicircle with center M. Two semi-circles are drawn with AM and MB as the diameters. A circle is drawn such that it is tangent to all the three semi-circles. If AM = \(r\) units, find the radius of the circle (in terms of \(r\)).
Attachment:
The attachment semi.PNG is no longer available


A) \(\frac{r}{2}\)
B) \(\frac{r}{3}\)
C) \(\frac{r}{4}\)
D) \(\frac{r}{{\sqrt{2}}}\)
E) \(\frac{2r}{5}\)


As per the figure, the 3 centers of the 3 inscribed circle form an isosceles trianges with sides 0.5r+r1, 0.5r+r1,r. find the length of altitude( which bisects the side with length r).

triangle FCD is right angled triangle

Or \((CD)^2 = (CG)^2 + (DG)^2\)
Or \((Alt)^2 = (r)^2 + (0.5r + r1)^2\)
Or \((Alt) = \sqrt{r1(r+r1)}\)

Make a equation for the outermost circle radius as
\(r= r1 + \sqrt{r(r+r1)}\)
Or \((r-r1)^2 = r(r+r1)\)
Or \(3*r*r1=r^2\)
Or \(r1 = (r/3)\)
Hence Option B

Cheers :)
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Pun91 Geometry.png [ 260.18 KiB | Viewed 303 times ]


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Originally posted by pun91 on 12 Jul 2018, 21:57.
Last edited by pun91 on 12 Jul 2018, 22:36, edited 1 time in total.
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Re: In the following figure, AB is the diameter of semicircle with center  [#permalink]

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New post 12 Jul 2018, 22:08
Princ wrote:
In the following figure, AB is the diameter of semicircle with center M. Two semi-circles are drawn with AM and MB as the diameters. A circle is drawn such that it is tangent to all the three semi-circles. If AM = \(r\) units, find the radius of the circle (in terms of \(r\)).
Attachment:
The attachment semi.PNG is no longer available


A) \(\frac{r}{2}\)
B) \(\frac{r}{3}\)
C) \(\frac{r}{4}\)
D) \(\frac{r}{{\sqrt{2}}}\)
E) \(\frac{2r}{5}\)


Please refer the affixed diagram,

OMD is a right angled triangle,
So, \(OD^2=OM^2+MD^2\)
Or, (\((r_{1}+(\frac{r}{2}))^2\)=\((r-r_{1})^2+(\frac{r}{2})^2\)
Or, \(rr_{1}=r^2-2rr_{1}\)
Or, \(r^2=3rr_{1}\)
Or, \(r=3r_{1}\)
Or, \(r_{1}=\frac{r}{3}\)

Ans. (B)
Attachments

Semicircle.JPG
Semicircle.JPG [ 22.36 KiB | Viewed 289 times ]


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Re: In the following figure, AB is the diameter of semicircle with center  [#permalink]

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New post 12 Jul 2018, 22:43
Thanks! This shows that even the seemingly complex questions in GMAT can be solved by simple and effective techniques like that of Pythagorus Theorem.

pun91 wrote:
Princ wrote:
In the following figure, AB is the diameter of semicircle with center M. Two semi-circles are drawn with AM and MB as the diameters. A circle is drawn such that it is tangent to all the three semi-circles. If AM = \(r\) units, find the radius of the circle (in terms of \(r\)).
Attachment:
semi.PNG


A) \(\frac{r}{2}\)
B) \(\frac{r}{3}\)
C) \(\frac{r}{4}\)
D) \(\frac{r}{{\sqrt{2}}}\)
E) \(\frac{2r}{5}\)


As per the figure, the 3 centers of the 3 inscribed circle form an isosceles trianges with sides 0.5r+r1, 0.5r+r1,r. find the length of altitude( which bisects the side with length r).

triangle FCD is right angled triangle

Or \((CD)^2 = (CG)^2 + (DG)^2\)
Or \((Alt)^2 = (r)^2 + (0.5r + r1)^2\)
Or \((Alt) = \sqrt{r1(r+r1)}\)

Make a equation for the outermost circle radius as
\(r= r1 + \sqrt{r(r+r1)}\)
Or \((r-r1)^2 = r(r+r1)\)
Or \(3*r*r1=r^2\)
Or \(r1 = (r/3)\)
Hence Option B

Cheers :)
_________________
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The fact that you aren't where you want to be, should be enough motivation.
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Re: In the following figure, AB is the diameter of semicircle with center &nbs [#permalink] 12 Jul 2018, 22:43
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In the following figure, AB is the diameter of semicircle with center

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