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) 100Ï€
B) 150Ï€
C) 200Ï€
D) 250Ï€
E) 300Ï€
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)
= 300Ï€
Answer: E
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
