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

*kudos for all correct

solutionsIf 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

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