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

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25 Jul 2015, 18:40

2

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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

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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Merged the posts.

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

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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

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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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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09 Nov 2016, 19:36

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Attached is a visual that should help.

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Screen Shot 2016-11-09 at 7.34.09 PM.png [ 241.73 KiB | Viewed 8871 times ]

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

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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

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

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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 )?

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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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

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

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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.

Attachments

hard gmat problem.png [ 313.1 KiB | Viewed 5862 times ]

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°

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

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

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

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

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