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# The two lines are tangent to the circle.

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GRE Instructor
Joined: 10 Apr 2015
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The two lines are tangent to the circle. [#permalink]  24 Jul 2018, 04:46
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Question Stats:

60% (02:54) correct 40% (02:12) wrong based on 10 sessions

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

_________________

Brent Hanneson – Creator of greenlighttestprep.com

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Joined: 28 Nov 2017
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Kudos [?]: 41 [0], given: 32

Re: The two lines are tangent to the circle. [#permalink]  24 Jul 2018, 10:04
GRE Instructor
Joined: 10 Apr 2015
Posts: 2387
Followers: 78

Kudos [?]: 2358 [3] , given: 33

Re: The two lines are tangent to the circle. [#permalink]  26 Jul 2018, 06:02
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Expert's post
GreenlightTestPrep 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

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

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com

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Joined: 19 Nov 2018
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Kudos [?]: 64 [1] , given: 45

Re: The two lines are tangent to the circle. [#permalink]  09 Sep 2019, 09:10
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Some people may find an introduction to, or refresher on, the Two Tangent Theorem and Radian Tangent Theorem to be helpful. Here is a succinct explanation I found: https://www.onlinemathlearning.com/tangent-circle.html
Re: The two lines are tangent to the circle.   [#permalink] 09 Sep 2019, 09:10
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