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Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
The two lines are tangent to the circle.
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Updated on: 08 Sep 2024, 00:14
Updated on: 08 Sep 2024, 00:14
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The two lines are tangent to the circle. If \(AC = 10\) and \(AB = 10 \sqrt{3}\), what is the area of the circle?
A) \(100 \pi\)
B) \(150 \pi\)
C) \(200 \pi\)
D) \(250 \pi\)
E) \(300 \pi\)
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Originally posted by GreenlightTestPrep on 24 Jul 2018, 04:46.
Last edited by Carcass on 08 Sep 2024, 00:14, edited 3 times in total.
Last edited by Carcass on 08 Sep 2024, 00:14, edited 3 times in total.
Updated
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
The two lines are tangent to the circle.
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26 Jul 2018, 06:02
26 Jul 2018, 06:02
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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
Answer: E
Cheers,
Brent
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