sureshbala wrote:Folks, finally here is the solution....Thought of being more clear on this problem, so uploaded the image....Have a look at this....
Very good question! It mixes up various concepts ranging from similarity of triangles to properties of different triangles- concepts that are frequently tested in the GMAT.
While the problem is solved efficiently by sureshbala,I've found the answer to the problem without using any sin, cosine, or whatnots. This way, I hope, the question can be perceived more relevant to the GMAT.
Here is how I did that:
As I don't know how to put figures in here, I want you to draw a figure for yourself and follow me along.
On the original figure, draw a perpendicular line form C to line AP, and let's call the intersection point N. Similarly, draw a perpendicular line from A to line CP, and let's call the intersection point M.
Since triangle CNP is 30-60-90 right triangle, and line CP, which is the hypotenuse, has a length of 2, line CN = Square root 3 and line NP = 1.
Now look at triangle AMB; it is a 45-45-90 right triangle. Thus, Length MB= Length AM. Let's represent length of MP by X; Therefore MB=AM =1+X.
Now look closer. Triangle AMP and triangle CNP are similar triangles. As such, their corresponding sides must be proportional.
[Length of NP/Length of MP] = [Length CN/ Length AM]
1/X = Square root (3)/ (1+X)
X = (Square root(3) + 1)/ 2
Now, we can get length of CM = 2-X = (3- Square root(3))/2
and AM = 1+X = (Square root(3) + 3)/2
Therefore, we can find length of AC by the pythagorean theorem
(AC)^2 = (AM)^2 + (CM)^2
= Square root(6)
Now look closer at the figure. Triangle ANC is right triangle; so we can get the length of AN by the pythagorean theorem.
So, length of AN = Square root(3)
See that? Triangle ANC has two equal legs; so it must be a 45-45-90 triangle.
Therefore, angle NCA =angle NAC = 45 degrees. Finally, as we are asked to get angle measure of BCA, we've to add 30 to 45 = 75 degrees.
