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# 380 from OG 10th Ed In the circle below, PQ is parallel

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Manager
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380 from OG 10th Ed In the circle below, PQ is parallel [#permalink]

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04 Apr 2006, 14:56
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#380 from OG 10th Ed

In the circle below, PQ is parallel to diameter OR, and OR has length 18. What is the length of minor arc PQ?

I need help with understanding one thing. In the answer explanation it says that "since the measure of inscribed angle PRO is 35 degrees, the measure of minor arc OP is 70 degrees." Can somebody explain to me how that is figured out? In other words, how do you figure out the 70 degrees by knowing that PRO is 35 degrees??

P.S. Sorry about my drawing skills...
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Manager
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04 Apr 2006, 15:19
Someone here probably knows the correct terminology better than I do, but in laymen's terms...any arc created from an angle that is started from the edge of a circle is equal to an arc eminating from the center of the circle when the angle in the center is 2 times the angle on the edge. Maybe in laymen's terms, it don't work so well. Gave it a shot.

BTW, is the answer to the question 9/8*pi?
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04 Apr 2006, 21:41
The key here is that minor arcs always have twice the measure of their insribed angles. Therefore, since PRO is 35, the minor arc OP has an angle of 70. Therefore QR is 70 as well. Now subtract 180 from 2*70 to find 40 (PQ). To find the length of the arc PQ, simply calculate the circumference and multiply it by 40/360 (or 1/9).
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05 Apr 2006, 06:50
Oops, made a dumb math error (180-140 is NOT 20, no matter what I think when doing this problem quickly).
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05 Apr 2006, 08:55
Thanks to both of you!

Just to clarify, given that the angle PRQ is 60 deg, does arc PQ in the below figure have an angle of 120 deg?
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Last edited by Matador on 05 Apr 2006, 15:49, edited 1 time in total.
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05 Apr 2006, 15:28
Thanks to both of you!

Just to clarify, does that mean that arc PQ in the below figure has an angle of 120 deg?

Matador: If the angle is at the centre is 60 then the arc is 120.
05 Apr 2006, 15:28
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