What is the length of AB if the radius of the circle with center O (sh

Question

https://gmatclub.com/forum/download/file.php?id=43495 What is the length of AB if the radius of the circle with center O (shown above) is 9 and the length of the arc ACD is 3π? A. 9\sqrt{3} B. 9\sqrt{2} C. 9 D. 3\sqrt{2} E. 3 image010.jpg

CounterSniper · Answer

length of an arc = 2 \pi r@/360 = 3 \pi 
@ = 60
which means triangle AOD = equilateral triangle 
AD=OD=9
using pythagoras
 (AB)^2  =  18^2  - 9^2  = 81(2)
AB= 9 \sqrt{3}

quantumdweller · Answer

What is the length of AB if the radius of the circle with center O (shown above) is 9 and the length of the arc ACD is 3π?

A. \(9\sqrt{3}\)

B. \(9\sqrt{2}\)

C. 9

D. \(3\sqrt{2}\)

E. 3

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Attachment:

image010.jpg

[/quote]

length of an arc = 2\(\pi\)r@/360 = 3\(\pi\)
@ = 60
which means triangle AOD = equilateral triangle
AD=OD=9
using pythagoras
\((AB)^2\) = \(9^2\) +\(9^2\) = 81(2)
AB= 9\(\sqrt{2}\)[/quote]

Apparently, Triangle OAB is assumed to be right angled (since largest side AB is considered as hypotenuse), which is not the case.
Triangle OAB is isosceles obtuse angle triangle, with angle OAB=120.

Since OA and OB and angle between them are known, one can directly compute AB by using vector algebra. Otherwise, one can prove that triangle ABD is right angles at angle DAB.
The answer is not B, but A.

Method1 (using geometry):

As calculated above @ or angle AOD=60 deg in triangle ADO

Since OD=OA=radius and angle AOD=60, hence ADO is equilateral triangle.

In triangle AOB, angle AOB= 180 - angle AOD

angle AOB=120
Also, OA=OB => angle OAB=angle OBA

By angle sum property applied at triangle AOB,
angle OAB=angle OBA=30

Now, angle DAB= angle OAD + angle OAB
=> angle DAB= 60 + 30 (triangle OAD is equilateral)
=> angle DAB=90

Hence, angle DAB is right angled triangle at angle A
Applying pythagoras theorem at triangle ADB

AB^2 + AD^2 = BD^2
AB^2= BD^2- AD^2

(BD=2*radius= 18; AD= 9)
Putting value,
AB= sqrt(243) = 9*sqrt(3)

CounterSniper · Answer

I took AB as the hypotenuse (sad mistake)

Corrected now

rahulkashyap · Answer

Cant this also be done without the info about the length of the arc?

We know that OB=OA=OD=r

Therefore, angle ADO = angle ABC= x

angle DAB = 90

x= 45

Therefore, 2 (AB)^2 = (18)^2

AB=9 root2

chetan2u  can you tell me whether the above reasoning is alright, mainly assuming that angle ADO = angle ABC= x

chetan2u · Answer

You have to know the length of arc, as other wise A could be anywhere and accordingly AB will vary...

As for your solution, it seems you have mixed up with variables..
But angle ADO=angle ABC or angle ABO if you meant ABO is not true

Solution..

Arc= 3π and circumference =2πr=2π*9=18π
So angle AOD =360*3π/18π=60. This in turn tells us that AOD is equilateral triangle...

Also ABD becomes 30-60-90 triangle 
So sides AD:AB:BD are in ratio 1:√3:2
If AD is r=9, AB =9√3

rahulkashyap · Answer

We know that OB=OA=OD=r

Therefore, angle ADO = angle ABO= x

angle DAB = 90

x= 45

Therefore, 2 (AB)^2 = (18)^2

AB=9 root2

can you let me know now where i am wrong?

chetan2u · Answer

\angle{ADO}=/angle {ABO}  means AD=AB
But is it given..NO

NidSha · Answer

Hi  chetan2u ,

I solved using the Rt. angle properties only - however the diff between your solution and mine, is that, the sides of the triangle corresponding to the angle based ratio don't match

My understanding says = for a rt. angle triangle with angles, 30-60-90 the sides length should be in ratio of 1:(sq root of 3):2 = where the side are the sides opposite to the angle of the triangle

hence, if in the mentioned ques - the angles are to be paired with their corresponding opp sides, they would be:

angle ADB = 30 degrees => corresponding side = AB
angle ABD = 60 degrees => corresponding side = AD
angle DAB = 90 degrees => corresponding side = DB

hence the ratio for 1:(sq root of 3):2 = AB:AD:DB = AB:AD:(9*2)

therefore, AB = DB/2 = 18/2 = 9; which is not the right answer

where am i wrong? pls guide

TIA

chetan2u · Answer

angle ADB is part of triangle ADO which is an equilateral triangle thus angle ODB=angle ADB = 60
but you have taken it as 30

abhinav5 · Answer

Angle in a semi circle is right angle.
Also 3pi/18pi=1/6 i.e Arc ACD will create 360 degree/6 at the center or 30 degree at circumference.
So the triangle is 60-30-90 and sides will be in the ratio root3:1:2
A) will be the answer.

NidSha · Answer

chetan2u

Thanks again for your quick response - actually i figured out my mistake later - in the "arc length to angle" calculation part, i made a mistake with treating angle ABD as 60 degrees instead of marking angle AOD as 60 degrees

Thanks again!

AsmaM · Answer

How I solved it was by imagining point A shifting to the right (straightening line OA) as if the triangle OAB was a right triangle, if that were the case then AB would be equal to 9 root 2, its not though and AB should be larger than that, fortunately within the answers there is no other option except for 9 root 3

bumpbot · Answer

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What is the length of AB if the radius of the circle with center O (sh

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What is the length of AB if the radius of the circle with center O (sh

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