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In the figure above, point O is the center of the circle and OC = AC =

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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 25 Jul 2015, 19:40
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DropBear wrote:

Geometry: What is the angle of x?



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In the figure above, point O is the center of the circle and OC = AC = AB. What is the value of x?

A. 40
B. 36
C. 34
D. 32
E. 30

I had to guess this one recently and even after reading the official answer and explanation there are still some inferences that I just don't understand. I am really looking forward to seeing a few different ways of solving. Personally this is one of the hardest questions I have faced... Maybe you will find it easy :)

The reason for the poll is I would like to see how difficult everyone finds this question. As I said, this one really beat me!

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Straightforward question if you realize the additional constraint would come from the fact that in Triangle AOB, OA = OB = radius of the circle.

Now in triangle, ACO , \(\angle{COA} = \angle{OAC} = x\) (as OC = AC) and \(\angle{ACB} = 2x\) (external angle of a triangle)

Additionally, \(\angle{ACB}= \angle{ABC}\) = 2x (as AC = AB)

FInally, in triangle AOB, OA = OB = radius of the circle ---> \(\angle{OBA}=\angle{OAB}\) = 2x ---> \(\angle{CAB} =x\)

Thus , in triangle ACB,

\(\angle{ACB} + \angle{CBA} + \angle{BAC}\) =2x+2x+x = 180 ---> x = 36. B is the correct answer.

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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 08 Sep 2015, 01:40
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the measure of the exterior angle is equal to the sum of the two non-adjacent angles of the triangle --> Angle ACB=2x
OA=OB => 180-4x+x=2x; X=36°
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 03 Oct 2015, 11:17
Something it's not quite right here.

I'm also stuck with the last calculation.

STEP 1
180-2x + y = 180
2x = y OK I KNOW HOW TO FIND THIS

STEP 2
Then I convert Y to become 2X
<AOB = x
<ABO = 2x
<BAO = x + (180-4x)-----Why I add x to 180-4x because that's how I think we get a straight line of 180 degree

therefore:
x + 2x + x + (180-4x) = 180
then I get 180 = 180

PLEASE HELP ME. TOTALLY STUCK
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 04 Oct 2015, 23:37
1
blendercroix wrote:
Something it's not quite right here.

I'm also stuck with the last calculation.

STEP 1
180-2x + y = 180
2x = y OK I KNOW HOW TO FIND THIS

STEP 2
Then I convert Y to become 2X
<AOB = x
<ABO = 2x
<BAO = x + (180-4x)-----Why I add x to 180-4x because that's how I think we get a straight line of 180 degree

therefore:
x + 2x + x + (180-4x) = 180
then I get 180 = 180

PLEASE HELP ME. TOTALLY STUCK


After step 2,
<BAO = <ABO
x + 180 - 4x = 2x
x = 36

The reason your third step doesn't work is because you used the property of total sum of triangle = 180 to get the relations. Now you are putting the relations back in sum of triangles is 180. You cannot get a value for x in this case. You need to put the relations in another property to arrive at a new conclusion (value of x).
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 09 Nov 2016, 20:36
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Attached is a visual that should help.
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 18 May 2017, 03:10
VeritasPrepKarishma wrote:
tonebeeze wrote:
In the figure attached, point O is the center of the circle and OC = AC = AB. What is the value of x (in degrees)?

a. 40
b. 36
c. 34
d. 32
e. 30


A small diagram helps:
Attachment:
Ques1.jpg


In the figure, you see (180 - 2x) + y = 180 (straight angle) so 2x = y

Also, the moment you see the circle and its two radii, mark them equal and the corresponding angles equal. (the red angle = blue angle)
x + 180 - 2y = y
5x = 180 (from above, y = 2x)
x = 36


good explanation but I am not understand (the red angle=blue angle )?
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 18 May 2017, 05:03
1
mkumar26 wrote:
VeritasPrepKarishma wrote:
tonebeeze wrote:
In the figure attached, point O is the center of the circle and OC = AC = AB. What is the value of x (in degrees)?

a. 40
b. 36
c. 34
d. 32
e. 30


A small diagram helps:
Attachment:
Ques1.jpg


In the figure, you see (180 - 2x) + y = 180 (straight angle) so 2x = y

Also, the moment you see the circle and its two radii, mark them equal and the corresponding angles equal. (the red angle = blue angle)
x + 180 - 2y = y
5x = 180 (from above, y = 2x)
x = 36


good explanation but I am not understand (the red angle=blue angle )?


OA and OB are radii of the same circle so they will be equal. So in triangle OAB, angle OAB = OBA ie. red angle = blue angle (in my diagram)
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 30 Jul 2017, 09:10
Very tough question. A key is recognizing that since OB is a radius and OA is a radius therefore angle OAB = angle OBA. The rest is also very tricky. Attached is a diagram.
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post Updated on: 16 Apr 2018, 12:27
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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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In the figure above, point O is the center of the circle and OC = AC = AB. What is the value of x ?

(A) 40
(B) 36
(C) 34
(0) 32
(E) 30


Since OC = AC, ∆AOC is an isosceles triangle, which means ∠OAC is also x°
Image

Since all 3 angles in ∆AOC must add to 180°, we can conclude that ∠OCA = (180-2x)°
Image

Since angles on a LINE must add to 180°, we can conclude that ∠ACB = 2x°
Image

Since AC = AB, ∆ACB is an isosceles triangle, which means ∠CBA is also 2x°
Image

Finally, since OA and OB are radii of the same circle, we know that ∆OAB is an isosceles triangle, which means ∠OABis also 2x°
Image

At this point, we can see that the 3 angles ∆OAB are x°, 2x° and 2x°
Since the angles in a triangle must add to 180°, we can write: x° + 2x° + 2x° = 180°
Simplify: 5x = 180
Solve: x = 36

Answer: B

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Originally posted by GMATPrepNow on 12 Oct 2017, 11:56.
Last edited by GMATPrepNow on 16 Apr 2018, 12:27, edited 1 time in total.
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 20 Oct 2017, 17:58
[quote="VeritasPrepKarishma"]
Hi,

if OC = AC = AB, then how come the three angles are not equal? (it's illustrated as x, y, and y in your diagram versus all y's)

Thanks
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 19 Mar 2018, 21:32
modernx wrote:
VeritasPrepKarishma wrote:
Hi,

if OC = AC = AB, then how come the three angles are not equal? (it's illustrated as x, y, and y in your diagram versus all y's)

Thanks



When two sides of a triangle are equal, the angles opposite them are equal. OC and AC form triangle OAC. The angles opposite to these two will be equal angle AOC = OAC. But we can't say what the measure of the equal angles is.

While AC and BC form triangle ABC. Angles opposite these will be equal angle BAC = CBA. But we can't say what the measure of the equal angles is.

Note that just because the triangles share a side AC, it doesn't mean the opposite angles will all be of the same measure. They are angles in different triangles. The length of the side does not give the angle.
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 28 Apr 2018, 05:20
VeritasPrepKarishma wrote:
tonebeeze wrote:
In the figure attached, point O is the center of the circle and OC = AC = AB. What is the value of x (in degrees)?

a. 40
b. 36
c. 34
d. 32
e. 30


A small diagram helps:
Attachment:
Ques1.jpg


In the figure, you see (180 - 2x) + y = 180 (straight angle) so 2x = y

Also, the moment you see the circle and its two radii, mark them equal and the corresponding angles equal. (the red angle = blue angle)
x + 180 - 2y = y
5x = 180 (from above, y = 2x)
x = 36



VeritasPrepKarishma hello :)

cant understand the logic behind this (180 - 2x) + y = 180 (straight angle) so 2x = y

if you are deducting 2x from 180 then why are you adding y ? there are 2x and 2y right and how 2x = y :?

why triangle OAB an isosceles and not an equilateral :?

please explain:)

have a great wekkend :)
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 06 May 2018, 08:53
1
dave13 wrote:
VeritasPrepKarishma wrote:
tonebeeze wrote:
In the figure attached, point O is the center of the circle and OC = AC = AB. What is the value of x (in degrees)?

a. 40
b. 36
c. 34
d. 32
e. 30


A small diagram helps:
Attachment:
Ques1.jpg


In the figure, you see (180 - 2x) + y = 180 (straight angle) so 2x = y

Also, the moment you see the circle and its two radii, mark them equal and the corresponding angles equal. (the red angle = blue angle)
x + 180 - 2y = y
5x = 180 (from above, y = 2x)
x = 36



VeritasPrepKarishma hello :)

cant understand the logic behind this (180 - 2x) + y = 180 (straight angle) so 2x = y

if you are deducting 2x from 180 then why are you adding y ? there are 2x and 2y right and how 2x = y :?

why triangle OAB an isosceles and not an equilateral :?

please explain:)

have a great wekkend :)


In triangle OAC, OC = AC
That is why angle COA = angle OAC = x
So angle OCA = 180 - x - x = 180 - 2x

Also AC = BC, so in triangle ACB,
angle ACB = angle CBA = y

Angles OCA and ACB form a straight angle so
180 - 2x + y = 180

In triangle OAB, OA = OB = radius of the circle. But these are not equal to AB (so not equilateral)
Note that AB is actually equal to OC (and AC). OC is less than the radius of the circle.
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Re: In the figure above, point O is the center of the circle and OC = AC =  [#permalink]

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New post 22 Dec 2018, 09:22
If you would like a video explanation

https://gmatquantum.com/official-guides ... nt-review/

From GMAT Quantum
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Re: In the figure above, point O is the center of the circle and OC = AC =   [#permalink] 22 Dec 2018, 09:22

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