In the given figure, O is the centre of the circle and if BAC = p - 1

Hi Experts KarishmaB, ThatDudeKnows, Bunuel,

Can you please share your approach to this question?

Thanks in advance.
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First things first, we can upfront eliminate options A and B as \(\angle BAC = p - 18\)

As \(\angle BAC = p - 18\), \(\angle BOC = 2(p - 18)\)

\(OD = BD\) ⇒ \(\angle BOD = \angle OBD = 2p - 36\)

\(BD = BC\) ⇒ \(\angle BCD = \angle DCB = x\)

\(\triangle OBD\)

\(2(2p-36) + 180 - x = 180\)

\(4p - x = 72\) ------ (1)

\(\triangle OBC\)

\(2x + 2p - 36 = 180\)

\(2p + 2x = 216\) ------ (2)

Using (1) and (2) we get

10p = 360

p = 36

Option D
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Assume p - 18 = x.

Now ∠BAC = x so ∠BOC = 2x (angle subtended at the centre is twice the angle subtended in the segment)
Because of equal sides of triangles, ∠OBD is also 2x. Then ∠BDC = 4x (sum of interior angles).
This means ∠DCB = 4x (angle opposite equal sides of a triangle)
So ∠CBD = 180 - 8x (sum of all angles in a triangle is 180 degree)

But OB = OC = radii of the circle so ∠OCB = ∠OBC i.e.

4x = 2x + 180 - 8x
x = 18 = p - 18
p = 36

Answer (D)
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I'm willing to bet that your gut reaction to my explanation is going to be dissatisfaction...that you're going to want to KNOW the right answer...to KNOW the math...to PROVE it...
But here's the thing: there are no points for any of that on the GMAT, only for getting the right answer, and ideally doing so as quickly as possible so that you can use any surplus time on other questions. So, as with the majority (yes, the majority of OG geometry questions of all difficulty levels) of geometry questions (at least until the new test curriculum comes into play), I suggest seeing if you can reason and ballpark your way to the right answer.

If p=16, p-18 is less than zero. Can't have that. Eliminate A.

If p=18, p-18 equals zero. Nope. Eliminate B.

Look at the figure. Really look at it. How are you going to get those three line segments to be equal? By having BAC be pretty small. Definitely smaller than 60-18=42. No way is that angle going to be almost 45 degrees. Eliminate E.

That took 8 seconds, and we are down to C and D.

If it's C, that would make BAC 6 degrees. If it's D, that would make BAC 18 degrees. Boy, doesn't 6 degrees just feel REALLY small? We know it's a small angle, but that's TOO small. Eliminate C. That took another 7 seconds.

Answer choice D.