Problem 9 · The circle through three points
A hidden right angle: $BC$ is a diameter.
Integer answer, at most 4 digitsA circle passes through the points $A = (2, 2)$, $B = (2, 4)$ and $C = (18, 2)$. Find the area enclosed by this circle. (The answer is a rational number times $\pi$; give that rational number as your answer. If it is a fraction, write the numerator and denominator of the irreducible fraction consecutively. For example, if the answer is $2\pi/3$, write 23, and if it is $50\pi$, write 50.)
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: $AB$ is vertical and $AC$ is horizontal, so the angle at $A$ is right. An inscribed $90°$ angle subtends a diameter: $BC$ is a diameter of the circle.
Compute $BC$ with the Pythagorean theorem:
$$BC^{2} = (18-2)^{2} + (2-4)^{2} = 256 + 4 = 260.$$
The radius satisfies $r = \dfrac{BC}{2}$, i.e. $r^{2} = \dfrac{260}{4} = 65$. The area is
$$\pi r^{2} = 65\pi.$$
Answer: 65