Geometry DS- GMATPrep

As shown in figure OB = OC = OD ( radius of the circle)
Since AB = OC ( given in the question)
=> AB = OB
and ABO will be isosceles triangle , with angle(BAO) = angle(BOA) = x

BOC is also an isosceles triangle, with angle(OBC) = angle(OCB) = x

Also, y = 2x ( Sum of interior opposite angles of a triangle)
=> x = y/2 So, if we know y we can find x

angle(BOA) + angle(BOC) + angle(COD) = 180 (straight line)
x + z + angle(COD) =180
Since z = 180 - 2y ( interior angles of triangle BOC)
x + (180 - 2y) + angle(COD) = 180
=> x + angle(COD) = 2y
=> angle(COD) = 4x - x (since y = 2x)
=> angle(COD) = 3x , x = angle(COD)/3

Hence, we can find the value of x if we know the value of y or the value of angle(COD).

Statement 1:
It provides the value of angle(COD)
Hence it is sufficient to find the value of x.

Statement 2:
It provides the value of y
It is also sufficient to find the value of x.

Hence each statement alone is sufficient.

The correct answer is D

In the figure shown, point O is the center of the semicircle and points B, C, and D lie on the semicircle. If the length of line segment AB is equal to the length of line sement OC, what is the degree measure of angle BAO?

(1) The degree measure of angle COD is 60.
(2) The degree measure of angle BCO is 40.

It is given that AB=OC.
Since OC and OB are both radii, OC=OB.
Thus:


EVALUATE THE EASIER STATEMENT FIRST.
Since statement 2 gives information about one of the equal angles, start with statement 2.

Statement 2: The degree measure of angle BCO is 40.
The result is the following combination of angles:

Thus, angle BAO = 20.
SUFFICIENT.

Statement 1: The degree measure of angle COD is 60.
In the combination of angles yielded by statement 2, angle COD = 60.
Thus, statement 1 implies the same combination of angles as does statement 2.
SUFFICIENT.

The correct answer is D.

Hi, Gentlemen,

I'm at lost here. I need help.

It said in the statement that AB = OC

That means the following
Since OC = radius then
AB = BO (since BO = radius)
AB = OC (since OC = radius)

Now, doesn't that mean...
<BAO = <BOA = <OBC = <BCO ?

What am I missing here?

Thanks in advance

Hi mdavidm_531

Concept: The angles corresponding to the equal sides of a triangle are equal.

consider triangle OBC
OB = OC = radius of the semi- circle
therefore, angle(OBC) = angle(OCB) ..... 1

consider triangle AOB
AB = OC (given)
=>AB = OB (from 1)
therefore, angle(BAO) = angle(BOA) ..... 2

Since we are considering sides of 2 different triangles, all the 4 angles need not be equal.
Hope it helps

Thanks, das.ashmita

I'm a bit convinced now.

What really ticks me though is BO.

It is a common side of both triangles.

So doesn't that mean <BAO = <BCO ?

Oh my I'm quite disappointed with myself that I don't get this.

Thing is, I made <BAO and <BCO equal

I was still able to determine that Statement (1) is sufficient.

However, my answer so Statement (1) is different from my answer that I could get from Statement (2). And just like the mantra in DS, the answers in DS questions don't contradict.

I know we aren't looking for the answer in Data Sufficiency questions. But I just want to know. I want to learn!

Hi David



The 2 angles must have a common base. In the figure above, <B & <C have a common base BC.

<BAO = <BCO will be equal only if OA = OC.
Here the common base is AC.

Hello Can someone pls explain how stmnt 1 is sufficient. thnks

Hi Anjali,

from the diagram, OC = OB (radius). This means 2 things:

a. AB = OB =OC(given in the question)
b. angle OBC = angle OCB. Assume both angles x

Therefore angle BOC = 180-2x

Assume angle BOA = y

Since OB = AB, then angle OBA = y

Therefore angle ABO = 180-2y

angle ABO + angle OBC = 180

180-2y+x=180

Therefore x=2y


The question asks us to find BAO which is y


Statement1: Angle COD is 60 degrees

angle AOB +angle BOC + angle COD = 180
y+180-2x+60 = 180
y-4y= -60

[spoiler]Therefore Y = 20. Sufficient[/spoiler]

Statement 2: Angle BCO = 40

We already know angle BCO =x and we have proved x=2y

[spoiler]Therefore y =x/2 = 20

Sufficient.
[/spoiler]

Therefore answer D

Regards
Anup