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gmatblood: Master | Next Rank: 500 Posts; Posts: 489; Joined: Tue Jul 05, 2011 11:10 amFollowed by: 5 members; Thanked: 28 times

by gmatblood » Sat Nov 05, 2011 9:23 am

Please help to answer this question.

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Source: Beat The GMAT — Problem Solving | Sat Nov 05, 2011 9:23 am

vaibhavgupta: Master | Next Rank: 500 Posts; Posts: 416; Joined: Tue Aug 30, 2011 2:18 pmFollowed by: 9 members; Location: Delhi, India; Thanked: 13 times

by vaibhavgupta » Sat Nov 05, 2011 9:31 am

gmatblood wrote:Please help to answer this question.

IMO B

OA?

If OA is A, IMO B
If OA is B, IMO C
If OA is C, IMO D
If OA is D, IMO E
If OA is E, IMO A

FML!! :/

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Sat Nov 05, 2011 9:47 am

Length of arc AXB = 2*(Length of arc BZC)
Length of arc AYC = 3*(Length of arc AXB) = 6*(Length of arc BZC)

Hence, circumference of the circle = (Length of arc AXB) + (Length of arc BZC) + (Length of arc AYC) = (1 + 2 + 6)*(Length of arc BZC) = 9*(Length of arc BZC) = (9/2)*(Length of arc AXB)

Now circumference of the circle subtends an angle of 360Â° in the center of the circle. Say, arc AXB subtends an angle of xÂ° in the center.

Then, (9/2)*(Length of arc AXB) subtends an angle of 360Â° in the center of the circle.
So, length of arc AXB will subtend an angle of (2/9)*360Â° = 80Â° in the center of the circle.

Now, measure of angle BCA = half of the measure of the angle subtended by arc AXB in the center of the circle = 40Â°

The correct answer is B.

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rijul007: Legendary Member; Posts: 588; Joined: Sun Oct 16, 2011 9:42 amFollowed by: 9 members; Location: New Delhi, India; Thanked: 130 times; GMAT Score:720

by rijul007 » Sat Nov 05, 2011 9:57 am

From the figure above :
Angle A = a
Angle B = b
Angle C = c

Circumference of thecircle = m

AXB = x = (2c/360) * m
BZC = z = (2a/360) * m
CYA = y = (2b/360) * m

x = 2z
2c = 2(2a)
c= 2a

y = 3x
2b = 3(2c)
b = 3c

2a + 2b + 2c = 360
a+b+c = 180
c/2 + 3c + c = 180
9c/2 = 180
c = 40

Option B

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gmatblood: Master | Next Rank: 500 Posts; Posts: 489; Joined: Tue Jul 05, 2011 11:10 amFollowed by: 5 members; Thanked: 28 times

by gmatblood » Sat Nov 05, 2011 10:28 am

Anurag@Gurome wrote:Length of arc AXB = 2*(Length of arc BZC)
Length of arc AYC = 3*(Length of arc AXB) = 6*(Length of arc BZC)

Hence, circumference of the circle = (Length of arc AXB) + (Length of arc BZC) + (Length of arc AYC) = (1 + 2 + 6)*(Length of arc BZC) = 9*(Length of arc BZC) = (9/2)*(Length of arc AXB)

Now circumference of the circle subtends an angle of 360Â° in the center of the circle. Say, arc AXB subtends an angle of xÂ° in the center.

Then, (9/2)*(Length of arc AXB) subtends an angle of 360Â° in the center of the circle.
So, length of arc AXB will subtend an angle of (2/9)*360Â° = 80Â° in the center of the circle.

Now, measure of angle BCA = half of the measure of the angle subtended by arc AXB in the center of the circle = 40Â°

The correct answer is B.

Could you please elaborate on the last part of the solution!

Thanks

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Sat Nov 05, 2011 10:40 am

There is a theorem which says, the angle at the center of a circle is twice the angle at the circumference if both angles stand on the same arc.

For proof, have a look at the middle of the following webpage with a heading "Angle at the Center" : https://www.mathsrevision.net/gcse/pages.php?page=13

Hence, measure of angle BCA = half of the measure of the angle subtended by arc AXB in the center of the circle as they both stand on the same arc, i.e. AXB

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