ΔABC, which is right-angled at B, is inscribed in a circle with centre

Question

ΔABC, which is right-angled at B, is inscribed in a circle with centre O and radius 6 units. If the length of the smaller arc between points A and B is 4π units, what is the length of line segment BC

A. 3
B. π
C. 3√3
D. 6
E. 2π

A. 3
B. π
C. 3√3
D. 6
E. 2π

nick1816 · Answer

AC is the diameter, length of AC=2*6=12
Circumference of the circle=2*pi*r=12*pi 
Angle AOB=(4pi/12pi)*360=120
Angle OAB=Angle OBA= (180-120)/2=30

AC/sin90=BC/sin30
BC=AC/2=12/2=6

briantoth6 · Answer

It is confusing that they say the "smaller arc" in reference to arc AB, because it is actually longer than arc BC.

aniketnsit90 · Answer

How can we say that AC is the diameter of the circle? 
Kindly share if I am missing some basic here.

vanam52923 · Answer

check this : https://www.teachoo.com/5521/1097/Angle ... /Theorems/ So angle subtended by diamter at circumference is always 90 and viceversa. Since angle B is 90(and point B is on circumference) , so it means the diameter subtended that angle.

TarPhi · Answer

Bunuel   nick1816

Why can't the answer be 2π?

Total circumference is 12π
Arc AC = 6π
Arc AB = 4π
So Arc BC = 12π - 6π - 4π = 2π

AnirudhaS · Answer

You are absolutely right. But the question asks for line segment BC , not Arc BC. Hope this clears your doubt.

Satyavani · Answer

couldn't solve the above problem. can anyone explain with detailed explanation, thank you.

GMATinsight · Answer

Satyavani  Please watch the video for detailed explanation

aniketnsit90  Watch the video to understand the property why AC should be diameter

briantoth6  Watch the video to understand what a smaller arc (minor arc) means

nick1816  GMAT doesn't expect or want us to use trigonometry

ANswer: Option D

https://www.youtube.com/watch?v=XjOIg1BchpY

nick1816 · Answer

GMATinsight  Brother i know. But if someone knows higher mathematics concepts, he/she can use it. It  can  save some time.

GMATinsight · Answer

Happy to know that you are aware of it.

There are many misguided souls who don't. being an expert, I like to apprise people of facts here. I am sure you will take my previous comment and effort in a positive spirit. Just another suggestion, try to minimize use of higher maths while you are practicing. It's a good advice. There are many genuine reasons which might not sound convincing and meritorious but the bottom line is GMAT should be taken as GMAC wants. and there is definitely some merit in the suggestion. All the best and happy preparation.

TarPhi · Answer

GMATinsight  - thanks for the video.

I solved it slightly differently. Once you find angle AOB = 120 degrees and ABO and OAB is 30 degrees each, all you need to do is -
Angle COB is 60, Angle OBC is 60 therefore the 3rd angle is 60. Hence it is an equilateral triangle. Hence BC is 6

GMATinsight · Answer

TarPhi That's a very good inference to make. Good method. I feel jealous already.

TarPhi · Answer

Haha cheers man

thanks again!

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ΔABC, which is right-angled at B, is inscribed in a circle with centre

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ΔABC, which is right-angled at B, is inscribed in a circle with centre

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