Geometry

[prachi18oct]

Please explain!

[GMATGuruNY]

Since OB and OD are radii, OB=OD, as illustrated by the BLUE SEGMENTS in the figure above.
Thus, ∠OBD = ∠BDO.
Since it is given then BD=DA -- as illustrated by the PINK SEGMENTS in the figure above -- ∠BDA = ∠BAO.

A radius drawn to a point of tangency forms a RIGHT ANGLE.
Thus, radius OB and tangent BA form a right angle, implying that ∠OBA = 90.

We can PLUG IN THE ANSWERS, which represent the value of ∠BAO.
When we satisfy the constraint that ∠BDA = ∠BAO and that ∠OBD = ∠BDO, the result must be that ∠OBA = 90.

D: 45
Since ∠BDA = ∠BAO and ∠OBD = ∠BDO -- and ∠BDA and ∠BDO must sum to 180 -- the following figure is yielded:

Here, ∠OBA = 90+45 = 135.
Eliminate D.

B: 30
Since ∠BDA = ∠BAO and ∠OBD = ∠BDO -- and ∠BDA and ∠BDO must sum to 180 -- the following figure is yielded:

Here, ∠OBA = 60+30 = 90.
Success!

The correct answer is B.

[misterholmes]

I wouldn't follow a messy string of calculations to solve this. How much time do we have for this anyway.

Better to just observe the basics:
1. We all know isosceles triangles have equal base angles. There are two isosceles triangles in this picture. Label the angels.
2. A tangent to a circle makes a 90 degree angle with the radius.
3. Combine these two ideas and you get one equation in one variable.

Regards,

Mike Miagi
www.google.com/+MikeMiagi

[MartyMurray]

Please explain!

The three angles of a triangle add up to 180 degrees.

Looking at triangle 0BA we know that because AB is tangent to the circle, ∠OBA = 90. So we have one of the three angles of triangle OBA.

So the other two angles of triangle OBA add up to 180 - 90 = 90, meaning ∠BAO + ∠BOA = 90.

Now, since ∠OBA = 90, ∠ABD + ∠OBD = 90. Great. Why is this great?

Because BD = DA, meaning triangle ABD is isosceles, and ∠ABD = ∠BAO So that means ∠BAO + ∠OBD = 90.

We already know that ∠BAO + ∠BOA = 90. Therefore ∠OBD =∠BOA.

Now, we're almost there because OB and OD are radii, forming an isosceles triangle. So we know that ∠OBD = ∠ODB. From the above we also know that ∠OBD =∠BOA. So all three angles of triangle OBD are the same and it's an equilateral triangle with all angles measuring 60 degrees.

So ∠BOD = 60 and ∠BAO = 90 - 60 = 30

Choose B.