The circle above has radius 8, and AD is parallel to BC. If the length

Question

https://gmatclub.com/forum/download/file.php?id=41115 The circle above has radius 8, and AD is parallel to BC. If the length of arc AYD is twice the length of arc BXC, what is the length of arc BXC? A. 2π B. 8π/3 C. 3π D. 4π E. 16π/3 circle %282%29.jpg

Princ · Accepted Answer

OA: B IMG_20180501_190732.jpg \angle DAC= \angle ACB= 45^{\circ} (Alternate interior angle) Arc BUA and Arc CVD subtend 45 ^{\circ} at circumference so they will subtend 90 ^{\circ} at center of circle Length of Arc BUA =Length of Arc CVD = \frac{90^{\circ}}{360^{\circ}} *2*\pi *8 = 4\pi Length of Arc BUA +Arc CVD + Arc BXC +Arc AYD = Total circumference 4\pi+4\pi+Arc BXC+2*Arc BXC = 2*\pi*8 3*Arc BXC= 16\pi-8\pi Arc BXC= \frac{8\pi}{3}

bunny12345 · Answer

Option B is answer.
Angle C is 45* as angle A is given and segment BC and AD are parallel.

Now, consider points C and D. When lines re drawn from these points and intersected at point A on the circle, they make an angle 45*. WHen using these 2 points lines are made that intersect at Centre O, they will make an angle that is double of angle A i.e. 90*.
Similarly, Lines drawn from points B and A make an Angle of 90* at centre too.

Its given in question that Arc AYD is twice of arc BXC. This means that if angle subtended by arc BXC at centre is "x" then angle subtended by arch AYD will be "2x".

Hence centre has 4 different sectors. These sectors make an angles of x*, 2x*, 90*, and 90*. Total of these angles is 360. From this we get x as 60*.

SO, Angle BOC (O being at the centre of circle) is 60*.
Length of Arc BXC = (2 X pie X radius X Angle subtended at centre)/360
 = (2 x pie x 8 x 60)/360
 = 8pie/3...hence option B

JeffTargetTestPrep · Answer

We see that since arc CD corresponds to a 45 degree inscribed angle (angle CAD), arc CD is twice the measure of angle CAD. The measure of arc CD is 90 degrees, thus its arc length is 90/360 = 1/4 of the circle. We also can conclude that angle BCA is also 45 degrees (since BC is parallel to AD) and hence arc BA is also 90/360 = 1/4 of the circle. Since arc AYD is twice the length of arc BXC, we can let arc BXC = x (where x represents the fraction the arc length of BXC as of the circumference of the circle) and thus arc AYD = 2x. We can create the equation:

1/4 + 1/4 + x + 2x = 1

3x = 1/2

x = 1/6

Thus, arc BXC is 1/6 of the circumference of the circle, and arc AYD is 1/3 of the circumference of the circle.

Since the radius is 8, the circumference is 16π and thus arc BXC is 1/6 x 16π = 16π/6 = 8π/3.

Answer: B

generis · Answer

circle %25282%2529ed.jpg The arc is a fraction of the circumference Find the arc length: (1) Find the lengths of the other two minor arcs Minor arc AB and minor arc CD ( pink in diagram) are intercepted by two 45° angles Angle CAD = 45° = Angle BCA Those two angles are alternate interior angles of parallel lines cut by transversal AC Both 45° angles are inscribed Each arc opposite those angles = 90° Minor arcs AB and CD each = 90° Inscribed angle theorem: the arc is twice the measure of the inscribed angle Add the arc measures: 90°+ 90° = 180° (2) Remaining total arc length possible? (360° - 180°) = 180° remain for the arcs AYD and BXC (3) Measure of arc BXC? AYD is twice the length of arc BXC Let arc BXC = x Then arc AYD = 2x From above, there are 180° remaining x + 2x = 180° 3x = 180° x = 60° = arc length of BXC (4) Arc BXC is what fraction of the circumference? \frac{ArcMeasure}{360°}=\frac{60°}{360°}=\frac{1}{6} The arc length is \frac{1}{6} of the circumference (5) Circumference: 2\pi r = 2*\pi*8 = 16\pi (6) Arc length of BXC, which is \frac{1}{6} of circumference: (16\pi*\frac{1}{6})=\frac{16\pi}{6}=\frac{8\pi}{3} Answer B

HisHo · Answer

Hi  Princ

How did you get that AC intersects BD at the center of the circle not at any other point?

Princ · Answer

hisho AC does not intersect with BD at the center of the circle. My drawing is the problem, please check below image. circle.PNG I wanted to draw this.

Kinshook · Answer

https://gmatclub.com/forum/download/file.php?id=41115 
Given: The circle above has radius 8, and AD is parallel to BC. 
Asked: If the length of arc AYD is twice the length of arc BXC, what is the length of arc BXC?

Since arc AB & CD extend angle 45 at the circumference & 90 at the centre, their length each = 90/360 * 2π * 8 = 4π

Let the length of arc BXC = x
the length of arc AYD = 2x

4π + 4π+ 2x + x = 2*π* 8 = 16π
3x = 8π
x = 8π/3

IMO B

Regor60 · Answer

Angle BCA is also 45 degrees due to alternate interior angles.

Since angles at the periphery subtend arcs twice their angle, together AB and CD are half the circumference, or:

8π

This leaves arcs BXC and AYD together to form the other half, also 8π, of the circle.

Since AYD is 2*BXC:

AYD + BXC = 2BXC+BXC= 8π or

3BXC = 8π and BXC = 8π/3

Posted from my mobile device

GMATGuruNY · Answer

Circumference = 2πr = 2*π*8 = 16π

Arc CD:
Since inscribed angle CAD that intercepts arc CD = 45 degrees, the central angle that intercepts arc CD = 90 degrees, implying that arc CD consititutes 1/4 of the circumference:
1/4 * 16π = 4π

Arc AB:
Since line segments AD and BC are parallel, arcs AB and CD are equal in length.
Thus:
arc AB = 4π

Arc BXC:
Since arcs CD and AB together constitute 8π of the circumference, arcs BXC and AYD must consitute the remaining 8π.
Since BXC:AYD = 1:2 -- a ratio that has 3 parts, 1 of the 3 parts attiributed to arc BXC, the other 2 parts attritubed to arc AYD -- arc BXC must consitute 1/3 of the remaining 8π:
1/3 * 8π =  \frac{8π}{3}

B

The circle above has radius 8, and AD is parallel to BC. If the length

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The circle above has radius 8, and AD is parallel to BC. If the length

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