In the circle above, CD is parallel to diameter AB and AB

Question

https://gmatclub.com/forum/download/file.php?id=19003 In the circle above, CD is parallel to diameter AB and AB has length 16. What is the length of arc DB? (A) 2π (B) 7π/3 (C) 2π/3 (D) 4π/3 (E) 8π/3 circle_ABCD.png

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=19003 In the circle above, CD is parallel to diameter AB and AB has length 16. What is the length of arc DB? (A) 2π (B) 7π3 (C) 2π3 (D) 4π3 (E) 8π3 Since CD is parallel to diameter AB, then in-the-circle-above-pq-is-parallel-to-diameter-or-93977.html

fozzzy · Answer

Any other way to solve this problem? if you didn't know the central angle theorem.

Bunuel · Answer

Central Angle Theorem is a must know for GMAT. Check more here: math-circles-87957.html

fozzzy · Answer

So I made changes to the diagram based on what I understood so <1 = 60, <2=30. Correct me If i'm wrong.

Bunuel · Answer

It's correct. Check here: math-circles-87957.html

fozzzy · Answer

I've added a modified image hope it helps and O is the center.

the portion in red is the arc with the respective angles

Angle COA = Angle DOB = 60 degrees

So the sum of the angles needs to be 180 degree so angle COD is 60 degree

after that apply the formula 2 pi R * 60 / 360 -----> we are given diameter 16 so then radius becomes 8

solve the equation we will get option E.

kanusha · Answer

Central angle/360 = arc/circumference
So An inscribed angle is exactly half the corresponding central angle.
Inscribed angle is 30, then central angle is 60 so E….
Circumfrence = 2pir , so r= d/2, then 16pi
So 8pi/3, E….

HMC · Answer

Learning for me : central angle is double the inscribed angle....

Sent from my Lenovo A7000-a using  GMAT Club Forum mobile app

michaelkalend · Answer

Hi Bunuel!

Would you be so kind and explain this point in more details please? 
Consider, O to be the center of the circle, then according to the central angle theorem above <BOD=2<BCD=60.

As I understand, the central angle has to be twice more than other inscribed angle. Please clarify. This point.

Bunuel · Answer

Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle, so the central angle is twice the inscribed angle. You can check for more here: https://gmatclub.com/forum/math-circles-87957.html

michaelkalend · Answer

Yes, that is absolutely clear! But how is that applied to the following:

Consider, O to be the center of the circle, then according to the central angle theorem above <BOD=2<BCD=60.

How did you arrive with <BOD and does it mean that angle BOD is 120, while angle BCD is half which means it equals 60? 
And if I am getting above correctly, how did you arrive that angle BOD is 120?

Please explain.

Bunuel · Answer

Check the image below: https://gmatclub.com/forum/download/file.php?id=35435 Blue (central angle) and Green (inscribed angle) subtend the same Red arc DB, thus (Blue angle) = 2*(Green angle). H ope it's clear now. Untitled.png

Leo8 · Answer

FullSizeRender (2).jpg Length of the arc CDB = 2 pi * 4 * \frac{120}{360} = \frac{8 pi}{3}

CounterSniper · Answer

CD and AB are parallel .
Which means angle DCB and angle ABC are alternate angles
since alternate angles are equal
angle.DCB=angle.ABC=30
also ,
angle subtended by an arc at the centre of a circle = 2 * angle subtended by the same arc on any other point  on the circle .

length of arc = 2pir@/360 (@= angle subtended by arc at the centre of the circle)
length of arc.BD=2*pi*8*60/360
8pi/3
E

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In the circle above, CD is parallel to diameter AB and AB

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In the circle above, CD is parallel to diameter AB and AB

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