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705-805 (Hard)|   Geometry|                              
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In the figure above, point O is the center of the circle and OC = AC = 18 May 2015, 11:19
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erikvm
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
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Hi erikvm,


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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In the figure above, point O is the center of the circle and OC = AC = 25 Jul 2015, 19:40
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DropBear

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!

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

Please search for a question before posting.

Merged the posts.
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In the figure above, point O is the center of the circle and OC = AC = 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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In the figure above, point O is the center of the circle and OC = AC = 03 Oct 2015, 11:17
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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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In the figure above, point O is the center of the circle and OC = AC = 04 Oct 2015, 23:37
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blendercroix
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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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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In the figure above, point O is the center of the circle and OC = AC = 09 Nov 2016, 20:36
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Attached is a visual that should help.
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In the figure above, point O is the center of the circle and OC = AC = 18 May 2017, 03:10
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tonebeeze
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
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good explanation but I am not understand (the red angle=blue angle )?
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In the figure above, point O is the center of the circle and OC = AC = 18 May 2017, 05:03
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VeritasPrepKarishma
tonebeeze
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 )?
Show more

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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In the figure above, point O is the center of the circle and OC = AC = 20 Oct 2017, 17:58
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[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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In the figure above, point O is the center of the circle and OC = AC = 19 Mar 2018, 21:32
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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
Show more


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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In the figure above, point O is the center of the circle and OC = AC = 28 Apr 2018, 05:20
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VeritasPrepKarishma
tonebeeze
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
Show more


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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In the figure above, point O is the center of the circle and OC = AC = 06 May 2018, 08:53
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dave13
VeritasPrepKarishma
tonebeeze
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 :)
Show more

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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In the figure above, point O is the center of the circle and OC = AC = 22 Dec 2018, 09:22
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If you would like a video explanation

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

From GMAT Quantum
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In the figure above, point O is the center of the circle and OC = AC = 07 Oct 2020, 04:55
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Bunuel
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°


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

Answer: B

RELATED VIDEO FROM OUR COURSE
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@brentprepnow cant we arrive at Angle ACB= 2x using external angle equals sum of opp interior angles?
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In the figure above, point O is the center of the circle and OC = AC = 07 Oct 2020, 06:16
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GMATaxe001


@brentprepnow cant we arrive at Angle ACB= 2x using external angle equals sum of opp interior angles?
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We can definitely use that property to find angle ACB.
That said, many/most students aren't aware of the external angle property. So I don't use it in my solutions.

Cheers,
Brent
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In the figure above, point O is the center of the circle and OC = AC = 07 Dec 2020, 21:54
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OFFICIAL GMAT EXPLANATION

Consider the figure above, where DB is a diameter of the circle with center O and AD is a chord. Since OC = AC, ΔOCA is isosceles and so the base angles, ∠AOC and ∠OAC, have the same degree measure. The measure of ∠AOC is given as x°, so the measure of ∠OAC is x°. Since AC = AB, ΔCAB is isosceles and so the base angles, ∠ACB and ∠ABC, have the same degree measure. The measure of each is marked as y°. Likewise, since OD and OA are radii of the circle, OD = OA, and ΔDOA is isosceles with base angles, ∠ADO and ∠DAO, each measuring z°. Each of the following statements is true:

i. The measure of ∠CAB is 180 − 2y since the sum of the measures of the angles of ΔCAB is 180.
ii. ∠DAB is a right angle (because DB is a diameter of the circle) and so z + x + (180 − 2y) = 90, or, equivalently, 2y − x − z = 90.
iii. z + 90 + y = 180 since the sum of the measures of the angles of right triangle ΔDAB is 180, or, equivalently, z = 90 − y.
iv. x = 2z because the measure of exterior angle ∠AOC to ΔAOD is the sum of the measures of the two opposite interior angles, ∠ODA and ∠OAD.
v. y = 2x because the measure of exterior angle ∠ACB to ΔOCA is the sum of the measures of the two opposite interior angles, ∠COA and ∠CAO.

Multiplying the final equation in (iii) by 2 gives 2z = 180 − 2y. But, x = 2z in (iv), so x = 180 − 2y. Finally, the sum of the measures of the angles of ΔCAB is 180 and so y + y + x = 180. Then from (v), 2x + 2x + x = 180, 5x = 180, and x = 36.
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In the figure above, point O is the center of the circle and OC = AC = 30 May 2021, 00:13
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Bunuel

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



Show Spoiler
Attachment:
The attachment Untitled.png is no longer available
Show more

Another approach:

Since, AC = AB, Angle ACB = ABC;
Also, since OC = AC, Angle AOC = Angle OAC = x

Angle ACB is the external Angle to Triangle AOC at C. Therefore, Ange ACB = 2(Angle AOC) = 2*x =2x
Since Angle ACB = Angle ABC, Angle ABC = 2x

Now extend the radius OB to M, making it the diameter (Illustrated in the attached image)

In Triangle MAB, Angle MAB = 90 (Angle in a semi circle)
Also, Angle AMB = x/2 (Angle subtended at the arc is half the angle subtended at the centre)
Also, for Triangle MAB,
Angle MAB + Angle AMB + Angle MBA = 180 (Interior angles of a triangle)
i.e. 90 + x/2 + 2x = 180
or, 5x/2 = 90
or, x = 36

Therefore, (B) is the correct answer
This is an indirect method. In Hindi this approach would be called "Olte haath se seedha kaan pakadna" (Holding right ear with left hand). But if it hits you at the right time, it works :)
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In the figure above, point O is the center of the circle and OC = AC = 06 Dec 2022, 17:21
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BrainLab
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°
Show more

KarishmaB

I found it to be more intuitive to have angle CAB = 180-4x as the user indicated above. However, I tried solving for x using the angles for the whole larger triangle combined so angles
COA + OAB + CBA
In other words, I did x + 180-4x + 2x + x and then set it equal to 180 because that is all that the whole triangle can equal. However, everything canceled out. Why doesn't this work? Thank you for your help :)
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In the figure above, point O is the center of the circle and OC = AC = 06 Dec 2022, 20:54
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BrainLab
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°

KarishmaB

I found it to be more intuitive to have angle CAB = 180-4x as the user indicated above. However, I tried solving for x using the angles for the whole larger triangle combined so angles
COA + OAB + CBA
In other words, I did x + 180-4x + 2x + x and then set it equal to 180 because that is all that the whole triangle can equal. However, everything canceled out. Why doesn't this work? Thank you for your help :)
Show more

Take a simpler example:
Say in a triangle, two angles are x and 2x. Now if we find that the third angle is also x, we can find the value of x using x + 2x + x = 180.
But if we ignore that and say that the third angle is (180 - 3x), can we use
x + 2x + (180 - 3x) = 180 ?
No, we have already utilised this property of sum of angles before.
In our original question too, the values are such that we can utilise this property once to get the value of x.
We must identify that angle CAB is also x and not use 180 - 4x.
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In the figure above, point O is the center of the circle and OC = AC = 25 Mar 2023, 07:00
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Bunuel

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



Show Spoiler
Attachment:
Untitled.png
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OC=AC
Hence,Angle AOC=x=Angle OAC
Angle ACB=Angle AOC + Angle OAC(As per the rule)=2x
AC=AB(Given)
Hence,Angle ACB=Angle ABC=2x
OA=OB(radius of the same circle)
Hence,AOB is an isosceles triangle.
Angle OAB=Angle OBA(2x)
Angle OAB =Angle OAC(x)+Angle CAB(180-4x)
Hence,
2x=x+180-4x
5x=180
x=36

Option B
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