Last visit was: 20 Jul 2024, 09:17 It is currently 20 Jul 2024, 09:17
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

# In the figure shown, if the side of the square is 40, what is the radi

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 94430
Own Kudos [?]: 642518 [25]
Given Kudos: 86706
GMAT Tutor
Joined: 27 Oct 2017
Posts: 1895
Own Kudos [?]: 5793 [8]
Given Kudos: 238
WE:General Management (Education)
General Discussion
Current Student
Joined: 18 Aug 2016
Posts: 531
Own Kudos [?]: 587 [2]
Given Kudos: 198
Concentration: Strategy, Technology
GMAT 1: 630 Q47 V29
GMAT 2: 740 Q51 V38
Manager
Joined: 23 Sep 2016
Posts: 185
Own Kudos [?]: 331 [2]
Given Kudos: 29
In the figure shown, if the side of the square is 40, what is the radi [#permalink]
1
Kudos
1
Bookmarks
Bunuel wrote:

In the figure shown, if the side of the square is 40, what is the radius of the smaller circles?

A. $$10(\sqrt{2} - 1)$$

B. $$20(\sqrt{2} - 1)$$

C. $$\frac{20(\sqrt{2} - 1)}{\sqrt{2}+1}$$

D. $$40(\sqrt{2} - 1)$$

E. $$\frac{40(\sqrt{2} - 1)}{\sqrt{2}+1}$$

Source: ExpersGlobal

Attachment:
Screenshot %2887%29.png

IMO C
AS the diameter of large circle is 40 as the side of square is also 40
then diagonal of square is $$40\sqrt{2}$$
and this diagonal will include 2 small circles and 1 large circle so radius of small circle is approx
$$\frac{(40\sqrt{2} - 40)}{2}$$( 40root2-40/2)
so $$20(\sqrt{2}-1)$$ (this will be a bit bigger than diameter of smaller circle)
we need radius then divide it by 2 which make bit bigger than radius as $$10(\sqrt{2}-1)$$

but as you can see both small circles are not touching in the corner of the square then the answer will be little less than the above answer
above answer in terms of value is 10*0.3=3 correct answer will be little less
A.10*0.3= 3 our answer were bith less than 3
B.6 way greater than 3 incorrect
C.6/2.3= 2.3 (bit less correct)
D.40*0.3= 12 (very large)
E.40*0.3/2.3= 5.3(greater than 3 incorrect)

-------------------------------------------------------------------------------------------------------------------------------

If you like my explanation than please give me KUDOS

Originally posted by rishabhmishra on 14 Mar 2018, 00:33.
Last edited by rishabhmishra on 26 Mar 2018, 23:54, edited 1 time in total.
Director
Joined: 16 Sep 2016
Status:It always seems impossible until it's done.
Posts: 641
Own Kudos [?]: 2143 [3]
Given Kudos: 174
GMAT 1: 740 Q50 V40
GMAT 2: 770 Q51 V42
Re: In the figure shown, if the side of the square is 40, what is the radi [#permalink]
1
Kudos
2
Bookmarks
Bunuel wrote:

In the figure shown, if the side of the square is 40, what is the radius of the smaller circles?

A. $$10(\sqrt{2} - 1)$$

B. $$20(\sqrt{2} - 1)$$

C. $$\frac{20(\sqrt{2} - 1)}{\sqrt{2}+1}$$

D. $$40(\sqrt{2} - 1)$$

E. $$\frac{40(\sqrt{2} - 1)}{\sqrt{2}+1}$$

Source: ExpersGlobal

Attachment:
The attachment Screenshot %2887%29.png is no longer available

IMO C.

This is a nice tricky question and I took over 4 mins to solve it. Everyone who have tried a solution have got a different answer and I am excited to find out the OA.

Please refer to the attached figure. We have to find t ( radius of the small circle ) as in the figure.

The figure is zoomed into the 1/4th part of the larger square.

The distance between one corner of the square and the circle is diagonal of the square of side 20 minus the radius of inscribed large circle of radius 20.
I have called the distance x. And x = $$20\sqrt{2} - 20$$

This is the trap option!

However this distance x is split between the radius of the circle t and and distance between the center of small circle and corner of large square t\sqrt{2}.

Hence by multiplying the x in the correct ratio we can find radius t.

t =$$x * 1/{\sqrt{2}+1}$$

t = $$\frac{20(\sqrt{2} - 1)}{\sqrt{2}+1}$$

Hence C.

Attachments

radius of circle.jpg [ 541.21 KiB | Viewed 8510 times ]

Intern
Joined: 14 Mar 2017
Posts: 3
Own Kudos [?]: 0 [0]
Given Kudos: 0
Re: In the figure shown, if the side of the square is 40, what is the radi [#permalink]
gmatbusters wrote:
OA should be C.
Side of the square = 40
Dia of big circle = 40.
Now diagonal of square = 40root 2,
Hence 40r*root2= 2*r*root2 + 2*r+40.

Solving we get r= 20(root2-1)/(root 2+1).

Shouldn't it 4r????
Intern
Joined: 14 Mar 2017
Posts: 3
Own Kudos [?]: 0 [0]
Given Kudos: 0
Re: In the figure shown, if the side of the square is 40, what is the radi [#permalink]
Dexter78424 wrote:
gmatbusters wrote:
OA should be C.
Side of the square = 40
Dia of big circle = 40.
Now diagonal of square = 40root 2,
Hence 40r*root2= 2*r*root2 + 2*r+40.

Solving we get r= 20(root2-1)/(root 2+1).

Shouldn't it 4r????

Sent from my ONEPLUS A5000 using GMAT Club Forum mobile app
Director
Joined: 16 Sep 2016
Status:It always seems impossible until it's done.
Posts: 641
Own Kudos [?]: 2143 [0]
Given Kudos: 174
GMAT 1: 740 Q50 V40
GMAT 2: 770 Q51 V42
Re: In the figure shown, if the side of the square is 40, what is the radi [#permalink]
Dexter78424 wrote:
gmatbusters wrote:
OA should be C.
Side of the square = 40
Dia of big circle = 40.
Now diagonal of square = 40root 2,
Hence 40r*root2= 2*r*root2 + 2*r+40.

Solving we get r= 20(root2-1)/(root 2+1).

Shouldn't it 4r????

Hi Dexter78424,

Not quite clear what you are asking exactly. But the trick part of this question is - the ability to zoom in onto 1/4th of the larger square and focus on that.

You can see the detailed explanation in my post above & also in many other posts. The OA is correct -> Option (C)

Please go through those and ask specific doubts if you have any..

Best,
VP
Joined: 10 Jul 2019
Posts: 1385
Own Kudos [?]: 577 [0]
Given Kudos: 1656
Re: In the figure shown, if the side of the square is 40, what is the radi [#permalink]
taking the approach of setting the (Diagonal of the Square) - (Diameter of the Circle) = 2 of the Diameters of the Smaller Circles will end up overvaluing the actual radius because the smaller circles do NOT touch the Vertices of the Square.

Rule 1: and Radius drawn to a Line Tangent to the Circle will be Perpendicular to the Tangent Line.

If you call the Radius of the Small Circle = little "r"

Drawing 2 radii = r ---- from the Center of the Smaller Circles to the Tangent Side of the Square will create a Square of side r in the Corner of the Larger Square.

Thus, the Diagonal from the Center of the Smaller Circle to the Vertex of the Square = r * sqrt(2) = Diagonal of Small created Square

the Distance from the Center of the Smaller Circle to the Point of Tangency at the Larger Circle = r

the Distance across the Center of the Inscribed Circle = Diameter = Side of Square = 40

thus:

Entire Diagonal of Large Square = r * sqrt(2) + r + 40 + r * sqrt(2)

40 * sqrt(2) = 2 * r * sqrt(2) + 2*r + 40

----DIVIDE both sides by 2----

20 * sqrt(2) - 20 = r * sqrt(2) + r

---take r Common---

20 * sqrt(2) - 20 = r * [ sqrt(2) + 1 ]

r = [20 * sqrt(2) - 20] / [1 + sqrt(2)]

-C-
Manager
Joined: 31 Jul 2018
Posts: 98
Own Kudos [?]: 15 [0]
Given Kudos: 76
Location: India
GMAT 1: 700 Q49 V36
GPA: 3
In the figure shown, if the side of the square is 40, what is the radi [#permalink]
Hi Bunuel
GMATGuruNY

Why doesn't it work to take the centroid formula 1/3 * 20(sqrt{2}-1)

I mean if you imagine the smaller circle as an incircle of an imaginary triangle with base as common tangent to both circles.

Then diameter of the smaller circle is perpendicular tangent and also angular bisector of square (coincides with the diagonal of the square)
Tutor
Joined: 04 Aug 2010
Posts: 1325
Own Kudos [?]: 3230 [2]
Given Kudos: 9
Schools:Dartmouth College
Re: In the figure shown, if the side of the square is 40, what is the radi [#permalink]
1
Kudos
Bunuel wrote:

In the figure shown, if the side of the square is 40, what is the radius of the smaller circles?

A. $$10(\sqrt{2} - 1)$$

B. $$20(\sqrt{2} - 1)$$

C. $$\frac{20(\sqrt{2} - 1)}{\sqrt{2}+1}$$

D. $$40(\sqrt{2} - 1)$$

E. $$\frac{40(\sqrt{2} - 1)}{\sqrt{2}+1}$$

Diameter of the large circle = side of the square = 40
Diagonal of the square $$= 40\sqrt{2}$$ ≈ 40*1.4 = 56
The following figure is yielded:
Attachment:

radius of smaller circle.png [ 184.7 KiB | Viewed 6456 times ]

Since each red line segment = 8, the diameter of each small circle = LESS THAN 8.
Thus, the radius of each small circle = LESS THAN 4.
Only C yields a value less than 4:
$$\frac{20(\sqrt{2} - 1)}{\sqrt{2}+1}$$ ≈ $$\frac{20(1.4-1)}{1.4+1} = \frac{8}{2.4} =$$ less than 4

Non-Human User
Joined: 09 Sep 2013
Posts: 34039
Own Kudos [?]: 853 [0]
Given Kudos: 0
Re: In the figure shown, if the side of the square is 40, what is the radi [#permalink]
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Re: In the figure shown, if the side of the square is 40, what is the radi [#permalink]
Moderator:
Math Expert
94430 posts