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GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2508
Given Kudos: 1053
GPA: 3.39
In the circle above, PQ is parallel to diameter OR, and OR has length
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22 Nov 2021, 06:49
22 Nov 2021, 06:49
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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. \(3\pi\)
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: In the circle above, PQ is parallel to diameter OR, and OR has length
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22 Nov 2021, 11:39
22 Nov 2021, 11:39
1
GeminiHeat wrote:
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. \(3\pi\)
First, since PQ is parallel to diameter OR, we know that arc PO = arc QR.
So, we can apply a circle property that says inscribed angles holding/containing arcs (or chords) of equal length must have the same angle measurement.
This means angle QOR must also be 35°
Next, we'll apply the following property:
The above property tells us that, if an inscribed angle and a central angle are holding/containing the same arc (or chord), then the central angle will be TWICE the inscribed angle
So, if we take our given diagram, and add a line from the center to point Q...
....then the central angle holding/containing arc QR must be 70°
We can apply the same logic to conclude....
...that the central angle holding/containing arc PO must be 70°
Finally, since angles on a line must add to 180°,
....the missing angle here is 40°
This means the length of minor arc PQ = 40/360 of the circle's circumference
The circumference of a circle = (pi)(diameter), and we're told that diameter OR has length 18.
So.....
The length of minor arc PQ = (40/360)(pi)(18)
= (1/9)(pi)(18)
= (18pi)/(9)
= 2pi
Answer: A
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General Discussion
Re: In the circle above, PQ is parallel to diameter OR, and OR has length
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22 Nov 2021, 11:21
22 Nov 2021, 11:21
Expert Reply
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.
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.
