A, B, C and D are points on the circumference of circle with center at

Question

https://gmatclub.com/forum/download/file.php?id=42331 A, B, C and D are points on the circumference of circle with center at O. AC is the diameter of the circle and OD is parallel to BC. If the area of the sector formed by radius OD and OC, shown by blue colour, is \frac{1}{6} th of area of circle, what is the measure of line DE? (A) \sqrt{3} (B) \sqrt{2} (C) 1 (D) \frac{\sqrt{3}}{2} (E) \frac{1}{2} New question self made CIRCLE1.png

dprchem · Answer

Hi,

What is the answer to this question?

I calculated as below:

Angle DOC = 60 since sector DOC is 1/6 of area of circle
Angle ABC =90 since angle made by diameter
Now extend DO to meet AB at F
Angle AOF=DOC=60
Since OD is parallel to BC, OF is also parallel to BC
Then triangle AOF and ABC are similar triangles and hence both are 30-60-90 triangles.
From this it can be deduced that AC=2 or radius =1 (calculated using BC=1)
Hence OD=1
Now Triangle ODE is also a 30-60-90 Triangle
Hence DE =sqrt(3)/2
Answer: D

chetan2u · Answer

Yes, you are correct. it is all about 30:60:90 triangle and parallel sides..
you can also solve without any further drawing..

Princ · Answer

OA: D 
 \angle COD = 60^{\circ}  as colored sector area is  \frac{1}{6}  of total circle's area
 \angle COD = \angle BCA = 60^{\circ}  (Alternate Opposite angles,  BC||OD )

Both  	riangle ACB  and  	riangle DOE  are  30^{\circ}-60^{\circ}-90^{\circ} 	riangle

In  	riangle ACB , Ratio of sides  BC:AB:AC  is  1:\sqrt{3}:2 .

As BC =1 , then  AC  would be 2 .

AC = 2 OD (Radius of circle),  OD = \frac{AC}{2} =\frac{2}{2} =1

In  	riangle DOE ,Ratio of sides  OE:ED:OD  is  1:\sqrt{3}:2 .

As  OD =1 ,  ED  would be  \frac{\sqrt{3}}{2}

Mansoor50 · Answer

triangles ODE and ABC are similar.
BC corresponds to DE
AC corresponds to OD

thus:

DE/BC equals OD/AC
DE/1 equals r/2r
DE equals 1/2.

where am i going wrong?

Princ · Answer

riangle  ODE and CAB are similar.
 \frac{OD}{AC}(Opposite \angle=90^{\circ})=\frac{DE}{AB}(Opposite \angle =60^{\circ})=\frac{OE}{BC}(Opposite \angle = 30^{\circ})

Mansoor50 · Answer

Thank u!

bumpbot · Answer

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A, B, C and D are points on the circumference of circle with center at

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A, B, C and D are points on the circumference of circle with center at

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