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# Arc XYZ lies on a circle centered at O. Is a < 55?

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Math Expert
Joined: 02 Sep 2009
Posts: 51280
Arc XYZ lies on a circle centered at O. Is a < 55?  [#permalink]

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27 Jan 2015, 06:40
2
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Difficulty:

45% (medium)

Question Stats:

68% (02:14) correct 32% (02:10) wrong based on 214 sessions

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Arc XYZ lies on a circle centered at O. Is a < 55?

(1) Angle ZOY = 35
(2) Angle XOY = 75

Kudos for a correct solution.

Attachment:

Q14_1.png [ 21.9 KiB | Viewed 3173 times ]

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Posts: 7112
Re: Arc XYZ lies on a circle centered at O. Is a < 55?  [#permalink]

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27 Jan 2015, 08:35
ans C...
both statements are required to get angle zox which is equal to 2a..
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Math Expert
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Arc XYZ lies on a circle centered at O. Is a < 55?  [#permalink]

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02 Feb 2015, 01:39
Bunuel wrote:

Arc XYZ lies on a circle centered at O. Is a < 55?

(1) Angle ZOY = 35
(2) Angle XOY = 75

Kudos for a correct solution.

Attachment:
The attachment Q14_1.png is no longer available

VERITAS PREP OFFICIAL SOLUTION:

The correct response is (C).

If we draw a radius from O to Y, we can fill in the two given values: ZOY = 35, and XOY = 75. OX and OZ are both radii of the same circle, so their opposite angles must be equal. We can use this information to solve for angle XYZ, and in term find the value for a.

To do the math, here’s how it would look:

Attachment:

Q14_2.png [ 24.07 KiB | Viewed 2730 times ]

180 = 35 + OYZ + OZY
145/2 = 72.5 = OYZ = OZY

180 = 75 + OXY + OYX
105/2 = 52.5 = OXY + OYX

Angle XYZ = 72.5 + 52.5 = 125

Angle a = 180 – 125
Angle a = 55

The answer to the question would always be “no.”

If you chose (A), this would allow us to solve for XYO, but we’d also need angle OYZ to find a.

If you chose (B), this would allow us to find OYZ, but we’d also need to know angle XOY to find angle XYO.

If you chose (D), both pieces of information are important, but neither is independently sufficient.

If you chose (E), you may have missed that XO = OY = OZ, since all radii in a circle must be equal, as well as the fact that angles opposite equal sides are also equal.
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Arc XYZ lies on a circle centered at O. Is a < 55?  [#permalink]

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02 Sep 2015, 08:03
Bunuel wrote:
Bunuel wrote:

Arc XYZ lies on a circle centered at O. Is a < 55?

(1) Angle ZOY = 35
(2) Angle XOY = 75

Kudos for a correct solution.

Attachment:
The attachment Q14_1.png is no longer available

The correct response is (C).

If we draw a radius from O to Y, we can fill in the two given values: ZOY = 35, and XOY = 75. OX and OZ are both radii of the same circle, so their opposite angles must be equal. We can use this information to solve for angle XYZ, and in term find the value for a.

To do the math, here’s how it would look:

Attachment:
The attachment Q14_2.png is no longer available

180 = 35 + OYZ + OZY
145/2 = 72.5 = OYZ = OZY

180 = 75 + OXY + OYX
105/2 = 52.5 = OXY + OYX

Angle XYZ = 72.5 + 52.5 = 125

Angle a = 180 – 125
Angle a = 55

The answer to the question would always be “no.”

If you chose (A), this would allow us to solve for XYO, but we’d also need angle OYZ to find a.

If you chose (B), this would allow us to find OYZ, but we’d also need to know angle XOY to find angle XYO.

If you chose (D), both pieces of information are important, but neither is independently sufficient.

If you chose (E), you may have missed that XO = OY = OZ, since all radii in a circle must be equal, as well as the fact that angles opposite equal sides are also equal.

Bunuel,

can the solution be like this ?

an inscribed angle is exactly half the corresponding central angle, so all inscribed angle that subtends to the same arc, are equal.
so - 75+35 = 110/2 = 55 ?
Attachments

File comment: example image

Capture.PNG [ 79.32 KiB | Viewed 2311 times ]

Intern
Joined: 28 Oct 2015
Posts: 22
Re: Arc XYZ lies on a circle centered at O. Is a < 55?  [#permalink]

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28 Oct 2015, 06:53
Hi there,

How do you guys know zox is 2a?

Thanks!
VP
Joined: 05 Mar 2015
Posts: 1003
Re: Arc XYZ lies on a circle centered at O. Is a < 55?  [#permalink]

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02 Aug 2016, 10:50
Bunuel wrote:

Arc XYZ lies on a circle centered at O. Is a < 55?

(1) Angle ZOY = 35
(2) Angle XOY = 75

Kudos for a correct solution.

Attachment:
The attachment Q14_1.png is no longer available

here is alternate solution
(1)ZOY=35 means YXZ=17.5 (central angle theorem)
not suff....
(2) similarly XOY=75 means XZY = 37.5
not suff...

combining both as angle a = sum of 2opp.internal angles of the triangle XYZ
so a=37.5+17.5----------->55
suff...

Ans C
Attachments

Q14_1.png [ 24.4 KiB | Viewed 1633 times ]

VP
Joined: 05 Mar 2015
Posts: 1003
Re: Arc XYZ lies on a circle centered at O. Is a < 55?  [#permalink]

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02 Aug 2016, 11:40
Bunuel wrote:

Arc XYZ lies on a circle centered at O. Is a < 55?

(1) Angle ZOY = 35
(2) Angle XOY = 75

Kudos for a correct solution.

Attachment:
The attachment Q14_1.png is no longer available

Ans is C as both internal angles of triangle XYZ is known and is nothing but sum of those angles =a=55

Ans C
Attachments

Q14_1.png [ 24.4 KiB | Viewed 1623 times ]

Intern
Joined: 26 Dec 2016
Posts: 29
Re: Arc XYZ lies on a circle centered at O. Is a < 55?  [#permalink]

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18 Oct 2017, 09:58
why are 145 and 105 divided by 2? Is it because they're inscribed angles?
Intern
Joined: 24 Oct 2016
Posts: 25
Re: Arc XYZ lies on a circle centered at O. Is a < 55?  [#permalink]

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23 Oct 2017, 07:31
Another way to solve this question would be to use the concept of central angle and inscribed angle i.e. inscribed angle is half of the central angle.
Re: Arc XYZ lies on a circle centered at O. Is a < 55? &nbs [#permalink] 23 Oct 2017, 07:31
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