AB is the diameter of the circle. CD is parallel to AB. What is the length of minor arc CD?
(1) The radius of the circle is 12.
(2) The measure of ∠CAB is 30º.
OA: [/img]c[/img]. Can someone explain?
GROCKIT - Circles DS
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- sl750
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To compute the length of the minor arc, we need the angle measure of the arc CD and the radius of the circle
Arc length CD/2*Pi*r = Angle measure of CD/360
Statement 1
Doesn't tell us what the angle measure of arc AD, BC, CD are. Insufficient
Statement 2
As CD is parallel to AB, and Angle CAB = 30, angle measure of arc BC is 60, similarly, angle measure of arc AD is 60. We know angle measure of AB is 180. Therefore angle measure of arc CD is 180-120=60
Still this is insufficient as we don't know the radius
Combining the two statements
CD/2*Pi*12 = 60/360. Sufficient
P.S
Inscribed angle is one half the central angle
Arc length CD/2*Pi*r = Angle measure of CD/360
Statement 1
Doesn't tell us what the angle measure of arc AD, BC, CD are. Insufficient
Statement 2
As CD is parallel to AB, and Angle CAB = 30, angle measure of arc BC is 60, similarly, angle measure of arc AD is 60. We know angle measure of AB is 180. Therefore angle measure of arc CD is 180-120=60
Still this is insufficient as we don't know the radius
Combining the two statements
CD/2*Pi*12 = 60/360. Sufficient
P.S
Inscribed angle is one half the central angle
- GMATGuruNY
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When the statements are combined:
When an inscribed angle (which is formed by 2 chords) and a central angle (which is formed by 2 radii) intercept the same arc, the inscribed angle = 1/2 the central angle.
Inscribed angle CAB and central angle COB both intercept arc BC.
Thus, COB = 60.
Since 60/360 = 1/6, the length of intercepted arc BC = 1/6 of the circumference.
Since CD is parallel to AB, inscribed angle DCA = 30.
Applying the same reasoning used above, we can determine that the length of arc AD = 1/6 of the circumference.
Since arc ADCB = 1/2 of the circumference, arc CD = 1/2 - 1/6 - 1/6 = 1/6 of the circumference.
Statement 1 indicates that r=12, implying that C = 24Ï€.
Thus, arc CD = (1/6)*24Ï€ = 4Ï€.
SUFFICIENT.
The correct answer is C.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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