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Length of the arc

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Length of the arc

by hai1 » Sun Mar 21, 2010 6:35 am
Can anyone point out where I am going wrong?

Question: In the circle in picture, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ?
A) 2 pi B)9 PI/4 C)7Pi/2 D) 9Pi/2 E) 3Pi

Ans: A

My approach: As angle ORP=angle RPQ; angle RPQ=35

Now length of the arc is (angle inside the arc/2Pi) times the circumference of the circle.

By this length of arc OP= (35/2 Pi)*18 Pi= 35 * 9
By this length of arc QR= (35/2 Pi)*18 Pi= 35 * 9

Hence length of arc PQ=9 Pi (length of semi circle)- (OP+QR)
35 is pi= 7 Pi/36
Hence OP +QR=7 Pi/2

thus PQ= 9 Pi-7Pi/2= 11 Pi/2.

Is my formula, "length of the arc is (angle inside the arc/2Pi) times the circumference of the circle" incorrect?
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by outreach » Sun Mar 21, 2010 7:11 am
as per inscribed angle theorem
if angle PRO = 35 then the arc OP = 70
if angle RPQ=35 then the arc OP = 70

arc OP+arc PQ+arc QR=180
arc PQ=40

length of arc=(angle/360)*2*pi*radius
=(40/360)*2*pi*9
= 2pi
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by kstv » Sun Mar 21, 2010 7:37 am
Though not stated in the Q stem in the diagram it is given that PRO is 35°
Let the centre be X in the figure given
If we find the angle PXQ the problem is solved.
The imp concept to know is that an chord subtends an angle which is twice the angel chord arc subtends at the circumference. ( I think this is broadly the concept but not sure of the wordings)
Step A- So if PRO is 35° , the chord is PQ . This chord PQ will subtend an angle = 2*35°= 70° at the centre so angle PXO will be 70°
Since PQ is parallel to OR angle QPR is = PRO = 35°
Same as Step A angle QXR = 2*QPR = 70°
So angle PXQ is 40°
360° = Pi * Diameter = 18 Pi
so 40° = 18Pi (40/360) =2 Pi
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by icemanjs4 » Mon Mar 22, 2010 1:44 pm
It might be worth mentioning that the diagram, as shown in your post is slightly misleading.
P and Q are both points on the circle. For PQ to be parallel to the diameter drawn, P has to be to the left (horizontally speaking) of the circle's center, while Q is to the right. In your diagram, P isn't actually on the circle.

The key takeaway here (for me) is that the minor arc on the left (OP) is the same as the minor arc on the right (QR)
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