Challenge Question: the two lines are tangent to the circle

Brent@GMATPrepNow wrote:

The two lines are tangent to the circle. If AC = 10 and AB = 10√3, what is the area of the circle?

A) 100π
B) 150π
C) 200π
D) 250π
E) 300π

Answer: E
Difficulty level: 650 - 700
Source: www.gmatprepnow.com

*I'll post a solution in 2 days

If AC = 10, then BC = 10


Since ABC is an isosceles triangle, the following gray line will create two right triangles...


Now focus on the following blue triangle. Its measurements have a lot in common with the BASE 30-60-90 special triangle


In fact, if we take the BASE 30-60-90 special triangle and multiply all sides by 5 we see that the sides are the same as the sides of the blue triangle.


So, we can now add in the 30-degree and 60-degree angles


Now add a point for the circle's center and draw a line to the point of tangency. The two lines will create a right triangle (circle property)


We can see that the missing angle is 60 degrees


Now create the following right triangle


We already know that one side has length 5√3


Since we have a 30-60-90 special triangle, we know that the hypotenuse is twice as long as the side opposite the 30-degree angle.

So, the hypotenuse must have length 10√3

In other words, the radius has length 10√3

What is the area of the circle?
Area = πr²
= π(10√3)²
= π(10√3)(10√3)
= 300π

Answer: E

Cheers,
Brent