The area of the circle passing through the po | vedclass

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(A) Let the points be $A(1, 2)$, $B(5, 2)$, and $C(5, -2)$.Plotting these points, we observe that $AB$ is a horizontal line segment of length $4$ and

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$8$

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(A) Let the points be $A(1, 2)$, $B(5, 2)$, and $C(5, -2)$.Plotting these points, we observe that $AB$ is a horizontal line segment of length $4$ and $BC$ is a vertical line segment of length $4$.Since $AB \perp BC$, the triangle $	riangle ABC$ is a right-angled triangle with the right angle at $B$.For a circle passing through the vertices of a right-angled triangle, the hypotenuse is the diameter of the circle.The hypotenuse is $AC = \sqrt{(5-1)^2 + (-2-2)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$.Thus, the diameter of the circle is $4\sqrt{2}$.The radius $r$ is half of the diameter, so $r = \frac{4\sqrt{2}}{2} = 2\sqrt{2}$.The area of the circle is $\pi r^2 = \pi(2\sqrt{2})^2 = \pi(8) = 8\pi$.

The area of the circle passing through the points $(5, 2), (5, -2),$ and $(1, 2)$ is (in $\pi$)

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