GROCKIT - Circles DS

[rahulvsd]

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?

[sl750]

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

[rahulvsd]

Hi
Can you explain how angle subtended by the arc is 60 degrees ? Am stuck with this part of the problem.

[GMATGuruNY]

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?

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.