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Geometry - Circles

by neelgandham » Mon Nov 28, 2011 7:08 am
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA? (Please refer to the attachment)

1)20
2)30
3)40
4)50
5)60
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by gmatclubmember » Mon Nov 28, 2011 7:17 am
The angles will be in the ratio of the arcs.
Lets say arc BC is x and arc AB 2x and arc AC 6x.
That would mean the ratio of the corresponding opposite angles would be 20,40,120 (x+2x+6x=180, so x=20).
Answer is 40.
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by pemdas » Mon Nov 28, 2011 7:53 am
AB=2BC, AC=3AB=6BC, the ratio is BC:AC:AC=1:2:6

180`=9x, x=20`
angle BCA=2x=40`
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by GMATGuruNY » Mon Nov 28, 2011 8:38 am
neelgandham wrote:The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA? (Please refer to the attachment)

1)20
2)30
3)40
4)50
5)60

Image
An inscribed angle is formed by two chords.
Thus, angle BCA is an inscribed angle.

AB is the arc intercepted by inscribed angle BCA.
The degree measurement of an inscribed angle = 1/2 * the degree measurement of the arc intercepted by the inscribed angle.
Thus, angle BCA = (1/2)(arc AB).

Let BC = 1.
Since AB is twice BC, AB = 2.
Since AC is three times AB, AC = 3*2 = 6.
Thus, AB/circumference = 2/(1+2+6) = 2/9.

Since the circle = 360 degrees, the degree measurement of AB = (2/9)360 = 80.
Thus, angle BCA = (1/2)(arc AB) = (1/2)80 = 40.

The correct answer is C.
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by SwatiAgarwal » Tue Nov 29, 2011 8:54 am
neelgandham wrote:The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA? (Please refer to the attachment)

1)20
2)30
3)40
4)50
5)60
This problem is a combination of "Ratio box" and geometry.
Tosolve quickly
1) make ratiobox and divide 360 in ratio of 1:2:6 == 40:80:240
2) use geomentry that angle at center is twice at periferi
=> required angle is 80/2 = 40
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