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. \(3\pi\)

The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle.

Let C be the center of the circle.

According to the central angle theorem above \(\angle PCO=2\angle PRO=70\).

As PQ is parallel to OR, then \(\angle QPR = \angle PRO=35\). Again, according to the central angle theorem above \(\angle QCR=2 \angle QPR=70\).

\(\angle PCQ=180-( \angle PCO+ \angle QCR)=180-70-70=40\).

Minor arc \(PQ=\frac{40}{360}*circumference=\frac{2\pi{r}}{9}=2\pi\)

Answer: A.

For more on circle check the circles chapter of Math Book (link in my signature).
_________________

40/180 = x/pi*18/2

x = 2pi

Answr - A
_________________

kp1811 can u explain this step
angle for arc PQ = 180 - ( 2arc QP )

how u got that ? plz update

we know arc OP = arc QR so we can write either 180 -(arc OP + arc QR) or 180 - 2OP or 180 - 2QR

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.
_________________

It is given in the stem: "PQ is parallel to diameter OR ...".
_________________

It should be \(\angle PCQ = 180 - (\angle PCO + \angle QCR) = 180 - 70 - 70 = 40\).
_________________

Bingo! central angle theorem is the rock star! Thanks for the valuable Math book Bunuel!

One other thing I missed in this problem is it asks for the minor arc length PQ. NOT length of PQ.

Follow up question. Is it possible to find length of the chord PQ with the information given in the question? If so what is the length of PQ?
_________________

Red central angle below is twice inscribed blue angle, because they subtend the same arc QR:



Hope it's clear.


_________________

Blue angle is an inscribed angle.
Red angle is a central angle.

The measure of inscribed angle is always half the measure of the central angle.

Check here for more: math-circles-87957.html
_________________