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The two lines are tangent to the circle.

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The two lines are tangent to the circle. [#permalink] New post 24 Jul 2018, 04:46
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Question Stats:

50% (02:09) correct 50% (01:41) wrong based on 4 sessions
Image

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
[Reveal] Spoiler: OA

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Brent Hanneson – Creator of greenlighttestprep.com
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Re: The two lines are tangent to the circle. [#permalink] New post 24 Jul 2018, 10:04
Explanation Please
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GRE Instructor
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Re: The two lines are tangent to the circle. [#permalink] New post 26 Jul 2018, 06:02
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GreenlightTestPrep wrote:
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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


If AC = 10, then BC = 10
Image

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

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

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.
Image

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

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)
Image

We can see that the missing angle is 60 degrees
Image

Now create the following right triangle
Image

We already know that one side has length 5√3
Image

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.
Image
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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Re: The two lines are tangent to the circle.   [#permalink] 26 Jul 2018, 06:02
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The two lines are tangent to the circle.

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