In the Figure Shown, Point O is the Center of the Semicircle and Points B, C, D Lie on the Semicircle

Question- 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 segment OC, what is the degree measure of angle BAO?

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

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.

“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 segment OC, what is the degree measure of angle BAO?”– this is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Approach 1:

Let us write down everything we know from the stem:

BO=CO=radius=AB
BO=CO=radius=AB --> triangles BOC and ABO are isosceles.
∠BAO=∠BOA and ∠BCO=∠CBO
∠CBO=2∗∠BAO

(1) The degree measure of angle COD is 60º:

∠BAO+∠ACO=∠COD=60º degrees (Using exterior angle theorem)
∠ACO=∠CBO=2∗∠BAO

So,∠BAO+∠ACO=2∗∠BAO+∠BOA=3∗∠BAO=60º
∠BAO=20º

Hence, the statement is Sufficient

(2) The degree measure of angle BCO is 40º:

∠BCO=40º --> ∠BCO=∠CBO=40º=2∗∠BAO
--> ∠BAO=20º

Hence, the statement is Sufficient
Since both the statements are sufficient, D is the correct answer.

Approach 2:

We can solve this question quickly if we do everything up front.
Let us look at the attached diagram

The portion marked in Red are equal => AB = OC ( given in the problem statement)

OB = OC ( radius)

Let Angle AOB = X

Statement 1 says COD = 60 = 3x ( as per diagram) => X=20
Hence, it is sufficient.
Statement 2 says BCO = 40 = 2x ( as per diagram) => X=20
This is also sufficient.

Since both are sufficient, D is the correct answer.

Approach 3:

We can solve the problem by considering the angle as “x”

Let angle BAO=x
As per the problem statement AB=BO
So, we have angle BOA=x

Angle CBO is an exterior angle to BAO
BOA it is equal to the sum of their individual angles
Hence, Angle CBO = x+x=2x

BO and CO are the two radius and hence they subtend equal angles thus BCO = 2x
and BOC = 180-4x
However, we need the value of x
Let us check the statements one by one:

Statement 1 gives COD
Since COD+BOC+AOB = 180
60+180-4x+x=180
We cans olve for x - Hence, it is sufficient

Statement 2 gives
BCO = 2x = 40
Since, we can calculate x hence it is also sufficient.

Correct Answer: D

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