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

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

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erikvm wrote:
What step am I doing wrong here? I can't figure it out. I don't understand how to go to 5x = 180

I just get to 4x + 180 - 4x = 180 aka 4x - 4x = 0..

edit: that last "8" kinda looks like a 5, but its supposed to be an 8
Image


Hi erikvm,

Image
The angle which you have written as 180 - 2y = x because OA = OB (radii of the circle). So ∠OAB = ∠OBA.

∠(OAC + CAB) = y = 2x
x + ∠CAB = 2x which gives us ∠CAB = x.

Now summing up the angles in triangle OAB, we get x + 2x + 2x = 180 i.e. x = 36

Hope this helps :)

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

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DropBear wrote:

Geometry: What is the angle of x?



Attachment:
The attachment Capture.PNG is no longer available


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!

Would be great to receive my first Kudos if you find this useful too :-D


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.

Please search for a question before posting.

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

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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 [#permalink]

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New post 03 Oct 2015, 10:17
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 [#permalink]

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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 [#permalink]

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New post 09 Feb 2016, 00:09
Here, since OC=CA, angle OAC=x. Hence angle ACB=2x and ABC=2x
If we extend BO to Q, then we get an inscribed angle AQB which is half the central angle x.
and the angle QAB=90
so, x/2+2x+90=180
5x/2=90
x=36
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Re: In the figure above, point O is the center of the circle and [#permalink]

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New post 18 Jul 2016, 09:36
OA=OB (radius)

Therefore angle oab = angle oba

oc=ac(given)
therefore angle aoc = angle cao

therefore, angle oac = x degrees

angle acb = 2x degrees (because exterior angle =sum of remote interior angles)

ac=ab
therefore angle abc = 2x degrees(isosceles triangle rule)

ao=pb therefore, angle oab = 2x(proved before that angle abo =2x --->isosceles triangle rule)

triangle aob =180 = 2x+2x+x= 5x=180
therefore x= 36

therefore answer is B
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Re: In the figure above, point O is the center of the circle and [#permalink]

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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 [#permalink]

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New post 20 Nov 2016, 07:43
Hi Engr2012,

I was about to post the same logic as you described.

what I like about this method is how easy you get to the answer.

For anyone else trying to answer this question and in search for optimization. Take a look at Engr2012 post and my graph.
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Re: In the figure above, point O is the center of the circle and [#permalink]

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New post 23 Mar 2017, 06:42
I loved this question!

take a look at the triangle properties: if 2 sides are equal-then angles too
so as oc=ac means that AOC=OAC
x or AOB=180-2y
as AC=AB -> ABC=ACB->CAB is x

this means that angles OAB is made up of x + x=y
so x=180-2(2x)->5x=180, x=36
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Re: In the figure above, point O is the center of the circle and [#permalink]

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New post 18 May 2017, 02:10
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 [#permalink]

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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 [#permalink]

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New post 25 May 2017, 10:35
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

Attachment:
Untitled.png
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

Problem Solving
Question: 162
Category: Arithmetic Statistics
Page: 83
Difficulty: 600


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We know that OB and OA are radii of the circle, so their lengths are equal. And since OC = AC, we can denote their respective angles with x. For the supplementary angle of AC, we can use y and determine that 180-2x + y = 180 or 2x = y.

We also know that angle OAB = y, so x + 180 - 2y = y and thus x + 180 - 4x = 2x

-3x + 180 = 2x
180 = 5x

x = 36.
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Re: In the figure above, point O is the center of the circle and [#permalink]

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New post 30 Jul 2017, 08:10
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 [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

Attachment:
Untitled.png
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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Re: In the figure above, point O is the center of the circle and [#permalink]

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New post 20 Oct 2017, 16:58
[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
Re: In the figure above, point O is the center of the circle and   [#permalink] 20 Oct 2017, 16:58

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