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The center of the circle is at point \((0, 6)\). If the distance between the two points where the circle intersects the x-axis is 16, what is the area of the circle?

A. \(36\pi\) B. \(45\pi\) C. \(64\pi\) D. \(81\pi\) E. \(100\pi\)

The center of the circle is at point \((0, 6)\). If the distance between the two points where the circle intersects the x-axis is 16, what is the area of the circle?

A. \(36\pi\) B. \(45\pi\) C. \(64\pi\) D. \(81\pi\) E. \(100\pi\)

The radius of the circle \(= \sqrt{6^2 + 8^2} = 10\). The area \(= \pi10^2 = 100\pi\).

I took the wrong approach with this question, would someone kindly clarify?

I had done a quick sketch of 0,6 on a graph and assumed that since the distance between the first point and where x meets the number line (x, 0) then the hypotenuse would be 16, and the radius would therefore be 16.

Where am I going wrong? I recognize that a 6-8-10 triangle can be formed from the side with a value of 6 however how does length 16 factor into this? Thanks

I took the wrong approach with this question, would someone kindly clarify?

I had done a quick sketch of 0,6 on a graph and assumed that since the distance between the first point and where x meets the number line (x, 0) then the hypotenuse would be 16, and the radius would therefore be 16.

Where am I going wrong? I recognize that a 6-8-10 triangle can be formed from the side with a value of 6 however how does length 16 factor into this? Thanks

The center of the circle is on y-axis, so it's symmetric around it. We know that the the distance between the two points where the circle intersects the x-axis is 16, hence half of it (8 units) must be to the left of 0 and the remaining half (another 8 units) to the right of 0.

As you can see the radius of the circle is a hypotenuse of a right triangle with the sides equal to 6 and 16/2=8 (6-8-10 right triangle), so radius=hypotenuse=10. The area is \(\pi{r^2}=100\pi\).

I think this is a high-quality question and I agree with explanation. Wow. This is a 700 level question certainly not 600 as classified by you Bunuel. Anyway terrific question in my view! Great work!