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

A. 2\pi

B. \frac{9\pi}{4}

C. \frac{7\pi}{2}

D. \frac{9\pi}{2}

E. 3\pi

I hope this helps. The explanation goes as:
-We know angle CBA=35
-AOB=diameter and hence angle ACB=90
-Thus in triangle ACB, angle CAB=55
-Now in triangle AOC, OA=OC=Radius and thus angle ACO=55 and angle AOC=70.
-Similarly we can find angle DOB=70
-Thus angle COD=180-70-70=40
And length of arc calculation is given in the image=2π

good answer by Bunuel b y Central Angle theorem

Correct answer A

arc OP = arc QR = (70 * 18 pi )/360 as angle OPR = angle QPR (35 degree)

angle for arc PQ = 180 - (2 arc QP) = 180 - 140 = 40

length of arc PR = (40 * 18pi)/360 = 2pi

There are 2-3 methods, lets see one of them:---
1]You are given that OR is diameter=18 (radius =9); therefore arc OR is a semicircle
the semicircle is composed of 3 arc--arc OP + arc PQ+ arc QR {arc PQ = 180- arc OP + arc QR}

2]the lines PQ & OR are parallel , therefore than alternate angles are congruent ( angle PRO = angle=QPR )
but these are inscribed angles and we need the central angles to compute the length of arc given by the formula :
{central angle (teta) /360 * 2*pi*radius}

3] Central angle of arc QR or arc OP =2 * inscribed angle (2*35= 70)
thus arc's QR + OP have central angles 70 +70= 140

4] thus, central angle for arc PQ =180-140=40 ;

length of arc PQ= 40/360 *2* pi* 9 = 2 pi

You may also employ back solving if you are comfortable with that for faster solution
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Thanks = Kudos. Kudos are always welcome

got it kp thanks n appreciated
not comforatble with arc questions ........

I encountered this problem on one of the tests. It was around my 12th or 13 question. I do admit that this question stumped me and I took some amount of time in solving this question.

I started off with the geometric approach but then quickly got tangled in it. I then shifted to approximating the area and using the process of elimination. I am explaining my approach even though it is unconventional and based only on the situation that I was in.

assumption pi=3.14

The radius is given as (OR/2)=9. Hence the circumference of the circle is 18pi.

Clearly looking at the figure, the length of arc PQ is much smaller than 1/4 the circumference.

18*pi=18*3.14~42+ (read it as something greater than 42)

Hence 1/4 of the circumference is ~ 10+ (read as something greater than 10). The length of Arc PQ is much smaller than 1/4 of the circumference as well.

Options C and D can be eliminated since C=~10.5 and D=~13.5. These can be clearly eliminated. Similarly Option E : 3pi ~ very close to 10 and hence I eliminated it.

After this it was purely on the figure given than I eliminated the other choices. The arc PQ looks to be more like closer to half of the 1/4 i.e. ~5+. I really have no justification as to how I deduced it. Just an observation so that I could guess the answer.

Option B [(9pi)/4] comes to be roughly 7- (read as less than 7)

Option A (2pi) comes to ~6.5- (read as less than 6.5) which is closer to mu initial estimate of being 5+.

Hence I chose A.

I am very well aware that on some other day it is very much possible to choose B and to get the question wrong. Also I am not advocating this method in any way

Its just that this was an educated guess which I guessed had 50% chance of being right.

The above explanations and diagramatic representations have helped me realize my mistake and should such a problem appear again I am dead sure of the mathematical way to solve it.
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My attempt to capture my B-School Journey in a Blog : tranquilnomadgmat.blogspot.com

There are no shortcuts to any place worth going.

Bunuel how do we know that OR is the diameter of the circle?

<PCO AND < QPR ALTERNATE ANGLES? how are those 2 angles equal?

< RPQ = 35 degress

we have 2 inscribed angles so they are 70 degrees

now take one portion of the semicircle and we get 180 - 70 - 70 = 40 degrees

40 / 360 = 1/9th

Minor arc = Pi D

1/9 * 18 Pi = 2 Pi