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A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three

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Bunuel
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A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   09 May 2019, 20:46
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A quadrilateral is inscribed in a circle of radius \(200\sqrt{2}\). Three of the sides of this quadrilateral have length 200. What is the length of the fourth side?

(A) 200

(B) \(200\sqrt{2}\)

(C) \(200\sqrt{3}\)

(D) \(300\sqrt{2}\)

(E) 500
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nick1816
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A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   Updated on: 18 Jul 2019, 15:47
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Let Angle(AOC)=x
sides AB=AC=BD= b =200 and radius, r= \(200\sqrt{2}\)
cos(x)= \(\frac{(AO^2 + OC^2 - AC^2)}{2* AO*AO}\)
cosx= \(\frac{2r^2-b^2}{2r^2}\)
cosx=\(1-(\frac{b^2}{2r^2})\), where b =200 and r= \(200\sqrt{2}\)
cosx=1-(1/4) = 3/4

Now we know angle subtended by equal arcs are equal
Hence Angle(AOC)=Angle(AOB)=Angle(BOD)=x
Then Angle(COD)= (2pi-3x)

We know cos(2pi-3x)= cos3x
cox3x=\(4cos^3 x - 3cosx\)
cox3x= \(4*(3/4)^3 - 3(3/4)\) = -(9/16)

Also, cos3x=\(\frac{(CO^2 + OD^2 - DC^2)}{2* CO*OD}\)
\(-(9/16)=1-(CD^2/2r^2)\)
(CD^2/2r^2)=25/16
CD=500

It took me 10 minutes to solve this tho. Bunuel Can you suggest other method do this problem fast?




Bunuel wrote:
A quadrilateral is inscribed in a circle of radius \(200\sqrt{2}\). Three of the sides of this quadrilateral have length 200. What is the length of the fourth side?

(A) 200

(B) \(200\sqrt{2}\)

(C) \(200\sqrt{3}\)

(D) \(300\sqrt{2}\)

(E) 500

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figure 3.png
figure 3.png [ 6.79 KiB | Viewed 3957 times ]


Originally posted by nick1816 on 09 May 2019, 21:42.
Last edited by nick1816 on 18 Jul 2019, 15:47, edited 1 time in total.
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A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   18 Jul 2019, 03:46
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Bunuel wrote:
A quadrilateral is inscribed in a circle of radius \(200\sqrt{2}\). Three of the sides of this quadrilateral have length 200. What is the length of the fourth side?

(A) 200

(B) \(200\sqrt{2}\)

(C) \(200\sqrt{3}\)

(D) \(300\sqrt{2}\)

(E) 500


Hi Bunuel can you please post the official explanation of this one. It seems to be really tough.

Can experts please help with this one. VeritasKarishma , chetan2u, EgmatQuantExpert
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Re: A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   25 Apr 2021, 12:05
Hi Bunuel & chetan2u please can you help with the solution here?
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sujoykrdatta
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A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   25 Apr 2021, 13:10
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Bunuel wrote:
A quadrilateral is inscribed in a circle of radius \(200\sqrt{2}\). Three of the sides of this quadrilateral have length 200. What is the length of the fourth side?

(A) 200

(B) \(200\sqrt{2}\)

(C) \(200\sqrt{3}\)

(D) \(300\sqrt{2}\)

(E) 500


Please check the solution below:

Attachment:
IMG_20210426_013928__01.jpg
IMG_20210426_013928__01.jpg [ 1.83 MiB | Viewed 2742 times ]


A few points: I HAVE NOT assumed that BC is parallel to AD
It follows from the information.
Angle BAD = Angle CDA = x⁰
(Angle at circumference is half the angle at the center for the same arc)
Also, Angle BCD = Half of Major angle BOD
=> BCD = 1/2 * (360-2x)
=> BCD = 180-x
Thus, angles CDA and BCD add up to 180⁰
So BC || AD
I used this to show triangles OXY and OBC similar
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Re: A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   25 Apr 2021, 13:23
sujoykrdatta : That was an excellent explanation using the most basic concepts! Somehow these concepts never rang a bell in my head while solving!
Any suggestions?
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Re: A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   25 Apr 2021, 14:06
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beeblebrox wrote:
sujoykrdatta : That was an excellent explanation using the most basic concepts! Somehow these concepts never rang a bell in my head while solving!
Any suggestions?


This was a tough one. So it's okay to not have got the method in one go.

What I try doing in these cases is draw the diagram.
I drew a rough circle - assume that radius 200*rt2 and approximated a length equal to 200 and took that as the chord
How to approximately mark 200?
200rt2 ~ 200*1.4 = 280. Take half = 140. Take half of the rest again= 70
Add to get almost 200 (you get 210; that's okay)
- Shown as I in the diagram below.

That gave me the diagram and I realised that the longer side will pass above the center on the same side as the other chords.

// Just for the sake of clarity, I have shown the error you get if you draw the 4th side on the other side of the center.
- Shown as II in the diagram below
(This is not required as a part of the solution) //

Attachment:
IMG_20210426_023459__01.jpg
IMG_20210426_023459__01.jpg [ 1.25 MiB | Viewed 2646 times ]


So, sketching often leads to better idea generation

Now, once the correct diagram is done, I tried finding the angle relationships

Whenever there is a circle and the center is involved, I think of
1. Perpendicular from center bisects chord - this can also be used to solve - but is too long
2. Angle at center is twice at center
3. Radii are always equal and hence lead to isosceles triangles

Here #2 worked :)
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Re: A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   25 Apr 2021, 14:40
sujoykrdatta, while I understand the concepts of similarity, I find it slightly challenging to apply.
In an exam-like time-bound situation identifying the first pair of similar triangles(the way you identified), I fathom for me it would be difficult.
In order to enhance my intuitiveness towards such questions, what should I do?
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Re: A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   25 Apr 2021, 23:04
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beeblebrox wrote:
sujoykrdatta, while I understand the concepts of similarity, I find it slightly challenging to apply.
In an exam-like time-bound situation identifying the first pair of similar triangles(the way you identified), I fathom for me it would be difficult.
In order to enhance my intuitiveness towards such questions, what should I do?



The only way to identify similar triangles is to
1. mark the angles (even if those are in terms of variables)
2. name the angles that are equal
3. decide the correct order
4. that will decide which 2 triangles are similar
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Harshjha001
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Re: A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink] New post   26 Apr 2021, 01:15
Is it a good idea to make a wild guess and skip such questions in actual GMAT in case we have absolutely no idea about how to deal it ?
Because i know i wont be able to do it within 2 mins.
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Re: A quadrilateral is inscribed in a circle of radius 200*2^(1/2). Three [#permalink]
26 Apr 2021, 01:15
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