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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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12 Jul 2018, 21:10
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Question Stats:

45% (02:44) correct 55% (01:48) wrong based on 11 sessions

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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 $$r$$).

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

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

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

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

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

Attachment:

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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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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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) $$\frac{2r}{5}$$

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 [ 22.36 KiB | Viewed 289 times ]

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Joined: 13 Feb 2017
Posts: 1
Re: In the following figure, AB is the diameter of semicircle with center  [#permalink]

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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
_________________
Please hit kudos, As it will encourage me to be more active at the forum.
The fact that you aren't where you want to be, should be enough motivation.
Re: In the following figure, AB is the diameter of semicircle with center &nbs [#permalink] 12 Jul 2018, 22:43
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