In a circle centered on point O with radius 8, diameter AC and poi

Question

https://gmatclub.com/forum/download/file.php?id=29459 In a circle centered on point O with radius 8, diameter AC and point B on the circle, pictured above, what is the length of line segment (AB)? (1) ∠AOB = 120° (2) OB = BC Geometry_Q16Image.jpg

unverifiedvoracity · Accepted Answer

1. m∠AOB= 120°. So, m∠OAB + m∠OBA = 60°. (As sum of all angles in a triangle in 180°). OA = OB = radius, so m∠OAB = m∠OBA = 30°. m∠ABC = 90° and this is a 30-60-90 triangle. As you know one side (diameter or AC), you should be able to calculate the other 2 sides. Sufficient.

2.OB=BC. BOC is an equilateral triangle. So, m∠BCO = 60 = m∠BCA. As ABC is a right triangle, you have a 30-60-90 triangle. As you know one side (diameter or AC), you should be able to calculate the other 2 sides. Sufficient.

Answer is D.

mbaboop · Answer

1) tells you that the triangle on left is 120 - 30 - 30, and triangle on right is 60-60-60, and big combined triangle is 30-60-90. Can determine side lenghts with this info.
Suff

2) Side length of triangle on right gives you same info as 1) above. 
Again, Suff

D.

hsbinfy · Answer

d

1)angle 120..from which u can find out that triangle BOC is equilateral as OB and OC are radii of circle .ABC is right angled triangle Apply pythagoras theorem.

2) Same here also.OB=BC. triangle BOC is equilateral and ABC is right angled .Apply pythagoras theorem.

adiagr · Answer

Statement 1  
∠AOB = 120°. This means ∠BOC = 60°

Also because OA = OB (radius). Triangle AOB is a isosceles triangle. Therefore ∠OAB = ∠OBA = 30°

because AC is the diameter and angle ABC is right angle, so angle BCO will be 60 degrees.

Now in Triangle BOC, angle BCO = ∠BOC = 60°. Therefore OBC will also be 60°
 
So the triangle BOC is equilateral. Therefore side BC = side BO = radius of circle.

We know the radius, so we know BO. We know the radius so we know diameter AC.

From Pythagorus theorem (applied on triangle ABC) we can get AB.
 
 Statement 1 is sufficient.

Statement 2

OB = BC

We straight away get value of BC = side BO = radius of circle.

We know the radius, so we know BO. We know the radius so we know diameter AC.

From Pythagorus theorem (applied on triangle ABC) we can get AB.
 
 Statement 2 is also sufficient.

D is the answer.

SamBoyle · Answer

I think this is a little harder than Medium you need to know a lot of properties at once.

1) When you have a diameter with a triangle, the diameter is the hypotenuse. Therefore, the opposite side ∠ABC = 90
2) We also know that OB is a radius because it goes from the center to a point on a circle. Any point of the radius going to the edge of the circle is also a radius. Therefore OB=8 and OC =8
3) From this we know that Triangle OBC is Isosceles with BO and OC being equal meaning that ∠B and angle ∠C share a value

From this good start we see the diagram:
1) if ∠AOB =120 then the other side of that line ∠COB bust be 60 cuz it's a straight line and a line adds to 180. Now if we know that is 60 we know (180-60)/2 is the ∠ of B and ∠C since it is isosceles. Therefore we know now that this triangle is equilateral
2) From this we can subscribe a point of 8 on BC. From this we know it's a 90 degree triangle with one leg of 8² * X²=16² and from this we can solve

Similarly from Prompt 2
Knowing OB=BC
1) We know that OB =Radius =8 therefore BC =8 and we already knew that OC =8 So again equilateral and therefore we can solve.

GoodnessShallPrevail · Answer

∠ABC = 90°
Per Right Triangle Altitude Theorem:
AB square = AO X AC
AO = OC = 8
AC = AO + OC = 16
AB square = 8 X 16 = 128
Hence, AB could be determined from the Q stem itself.

So, irrespective of what information is given in each Statement, AB will be determined.
Hence, answer is D even before reviewing Statement 1 and Statement 2.

Please suggest whether the aforementioned is correct or not.

mihir0710 · Answer

My friend sorry to break it to you but above highlighted are a couple of incorrect statement in your method.

I am not sure from where did you get this Right Triangle Altitude Theorem because, in actual Altitude Theorem, the line segment BO has to have an angle of 90 Degree with line segment AC. (that is how you can call BO an altitude of the triangle.)

Hence, you are getting a wrong calculation.

Here is how I would approach this :

We are asked to find AB.
Now, Triangle ABC is a right angle triangle. 
So, AC^2 = AB^2 + BC^2
i.e AB^2 = AC^2 - BC^2

Now we know AC = 8 (twice the radius OC or OA or BO)
And if we know BC, we can find AB.
Hence, effectively, this question is to find BC.

Moving to the statements, 
St-1 : Angle AOB = 120.
So, In triangle AOB, Since AO=OB (both are radius), Angle OAB = Angle OBA
So from, Angle OAB + Angle OBA + Angle AOB = 180
2*(Angle OBA) = 180-120 = 60 
So angle OBA = 30
Therefore, Angle OBC = 90-30 = 60.

Hence, in the triangle, OBC, all angles are 60. 
Therefore, its an equilateral triangle and all sides have length 8 (same as radius OC).
So, BC = 8.

Hence this statement is sufficient.

Statement-2 : 
OB = BC
Pretty straight forward, OB = radius = 8 = BC
Hence, we got BC and so can calculate AB.

So, this statement is sufficient as well.

OPtion D is the correct answer.

P.S : If you were to calculate the value of AB, it would be 8Sqrt3. (which is NOT same as what you were calculating).

abhishekdadarwal2009 · Answer

well i see a lot op answers already posted so it might be redundent to post one more,but id like to post my way of solving the above proble.

So we know 
1.that all readius are equal
2.Angle made by a triangle with diameter as one of the side will always be 90 degrees(most important point,if you know this then consider problem solved)

Basically DS points 1 and 2 are both saying the same thing...that OC and OB are eqaul and if that is true then we get two sides of a right triangle and to find AB we simply use pythagorus theorm.

Answer is C.

dcummins · Answer

We need the following properties of triangles inscribed in circles to solve:
- if the longest side of an inscribed triangle is also the diameter of the circle in which it is inscribed then that triangle is a right triangle. Thus <ABC = 90
- Pythag rules- once we get 2 sides, we can find the 3rd side.

Statement 1.

Refer to diagram.

From the info, we can split triangle <ABO into two triangles by drawing a line from O to AB that is perpendicular (intersects at 90 deg); thus, allowing us to deduce the other angles.

Statement 2.

See Diagram 2

Once we get two sides

Bhavnapriya · Answer

Hi,
The questions say O is the radius, Therefore, <Abc=90degrees? Because any triangle which lies inside a semicircle, is a right-angled triangle?
Is this a flawed question?

jabhatta2 · Answer

chetan2u ,  Bunuel ,  VeritasKarishma ,  gmatbusters

Just reading the question stem -- why can't I assume all 4 angles are equal ?

Angle BAO = Angle BCO = Angle ABO = Angle OBC

My understanding is if 2 sides are equal, then their opposite angles are equal

Given the following three sides are equal (AO is equal to OC which is equal to OB) -- then their opposite angles to all these sides will be equal as well

chetan2u · Answer

Equal angles opposite the equal sides is for a particular triangle and cannot be taken as a rule when you have equal sides in two different triangles. 
So bao=abo and bco=cbo but there can be no definite relation between the two pairs.

Basshead · Answer

If AC is the diameter, and OC = 8, then AO = 8. We can conclude AC = 16. We can also conclude BO = 8, as BO is the radius.

What is the length of AB?

(1) ∠AOB = 120°

if AOB = 120, then BOC = 60. We can conclude BC = 8, meaning triangle BOC is an equilateral triangle. Since we know AC and BC, we can use the Pythagorean theorem to determine AB. SUFFICIENT.

(2) OB = BC

This statement tells us the same information provided in statement 1 (if you're ever provided the same information in both statements, the answer must be D or E)

BC = 8, and we know AC = 16. We can determine AB using Pythagorean theorem. SUFFICIENT.

Answer is D.

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In a circle centered on point O with radius 8, diameter AC and poi

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