In the figure above, the circles with centers A and B are tangent to

Question

https://gmatclub.com/forum/download/file.php?id=38932 In the figure above, the circles with centers A and B are tangent to each other at C, and are tangents to the lines m and n at F and D. If the radius of each of the circles is √2 and m is parallel to n, what is the length of EF? (A) 2√2 + 1 (B) 2√2 + 2 (C) 2√2 + 3 (D) 3√2 (E) 3√2 + 1 2017-11-30_1000.png

Sasindran · Answer

Bunuel I have a doubt. Without knowing the angle between the line AB and EF how to calculate the distance between ED? Please help.

Please refer the attached figure. For each case the length will be different.

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sohrabbanker · Answer

Hi, I tried an overly simplistic approach, and got the answer E.
It took me 10 seconds after trying all the theories I knew. Basically EF consists of 3 radi's plus something in the middle , so it must be 3√2 + 1. ( assuming the line opposite the radius is parallel to the radius.

Many assumptions taken, still not a concrete answer.

niks18 · Answer

Hi septwibowo Refer below image. I have joined points G & C where line GC will be common tangent to both the circles geo.jpg BC & AC will be perpendicular to line GC (radius is perpendicular to the tangent) So Triangles ACG & BCG will be similar (AC=BC=radius; GC is common side, angle GCB=angle GCA=90) => \frac{AC}{BC}=\frac{AG}{BG} =>AG=BG this means that triangle ABG will be isosceles right angle triangle, so sides will be in the ration 1:1:\sqrt{2} AB= \sqrt{2}+\sqrt{2}=2\sqrt{2} Therefore BG=2 Hence EF= EG+BG+BF EG=AD=\sqrt{2} So EF=\sqrt{2}+2+\sqrt{2}=2\sqrt{2}+2 Option B

Krystallized · Answer

Hi  niks18 ,

How did you conclude that a line tangent to both the circles from point C will meet at point G??

niks18 · Answer

Hi  Krystallized

Few thing to note here -
1. Point C is the point of tang-ency of both the circles
2. Radius is perpendicular to the tangent, so AC & BC will be perpendicular to the line passing through point C
3. Point C is also the mid-point of hypotenuse of right triangle AGB, where G is vertex
4. Perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.

You can always join the mid-point of the hypotenuse with the vertex. I do not see any issue here.

Is there any specific query you have in mind?

Krystallized · Answer

Hi  niks18

Thank you for your reply and explaination

I totally agree with the points mentioned by you in the above post.
My specific query is, how did you decide that a line perpendicular to AB from point C will intersect line BE at G?
I think this is possible only if angle ABG and angle BAG are each 45 degree or AG and BG are of same length. Otherwise the line would intersect BE somewhere below point G.
I know we can always join the midpoint of hypotenuse with the vertex, but that does not guarantee that the line joining those two points will always be perpendicular to the hypotenuse.

niks18 · Answer

Hi  Krystallized

I hope you would agree that triangle ABG is a right angled triangle with angle AGB being 90. AB will be the hypotenuse of this triangle with point C as its mid point. Point C is the point of tang-ency of both the circles so any tangent passing through point C will be perpendicular to AB (AC & BC are radius and radius is perpendicular to the tangent)

Next if from point G I draw a perpendicular bisector or simply a bisector of the hypotenuse AB, then Can you tell from which point the line will pass?

angle ABG and angle BAG will eventually each be 45 because Perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle and in my solution above it is evident that AG=BG i.e the it is an isosceles right angle triangle.

Also note, you cannot squeeze or expand the circles because radius is fixed. You cannot increase the distance between the parallel lines as it is also fixed. So if you are confused that such a situation can also arise at some other position by moving the circles, then that is not possible because in that case either the parallel lines will not be tangent or their distance will have to be increased/decreased. The figure is drawn in such a way that triangle ABG will be right isosceles, there is no other possibility otherwise you will have to alter the distance or the radius. you cannot move the upper circle lower otherwise it will not touch the upper tangent line and you cannot move the circle further high becuase in that case for upper line to be tangent you will have to squeeze i.e reduce the radius of upper circle.

DharLog · Answer

Excuse me, Niks18! Maybe I do not understand something, but I agree with Crystalized.
You say, we can not touch the circles. But I can put one of them exactly below another. And I will get an answer 4sqrt(2) (not in the options). Which initial conditions will I break?

In the figure above, the circles with centers A and B are tangent to each other at C - done
 and are tangents to the lines m and n at F and D - done 
If the radius of each of the circles is √2 and m is parallel to n, what is the length of EF? (EF wil be the same line DF)

niks18 · Answer

Hi DharLog Pls refer the image below - geo.jpg Triangle EFD is a right angled triangle with DF as its hypotenuse. The Hypotenuse is the LARGEST side of the triangle and it cannot be equal to its height EF, hence you do not see the option 4\sqrt{2} if you move the two circles either to the right or to the left then you will have to move the parallel line m & n as well to satisfy all the conditions given in the question. In the existing structure no movement of the circles is possible, hence you cannot put one circle exactly below the other

sumit411 · Answer

Bunuel

What is wrong with this approach?

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niks18 · Answer

Hi  sumit411 ,

how can you be certain that the "something in middle" is 1 and not 0.5 or 0.4 or any other value?

if "something in middle" is 0.414, then the answer will be B

Option B = 4.828427125 and

Option E = 5.242640687, between Option B & E there is a difference of only 0.414213562 and this difference cannot be predicted by naked eye

Anshkarha · Answer

Mate I don't understand how this has a unique answer. we're asked to find the distance between m & n so which is an unknown that relies on the position of the circles. There isn't any constraint given that locks the circles at a particular spot.

The constraints as per the question
i) Circles are tangent at C
ii) lines m & f are parallel at points F and D
iii) radius is rt2
iv) m is parallel to n

If I move the circles left and right, essentially moving point C, which of these constraints is violated? moving the parallel lines up and down doesn't change any of the above 4 points. 
Experts pls advise!

elPatron434 · Answer

chetan2u   Bunuel  could you please help us understand this better?

chetan2u · Answer

Hi 
I believe it is missing the information that ∆BAO is isosceles triangle that is angle BAO is 45°
If it is given than BA is 2√2 and BO=AO=2
EF=EB+BO+OE=√2+2+√2=2√2+2

Bunuel , would you please check the question. Even the choices are not in order..

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In the figure above, the circles with centers A and B are tangent to

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