Segment $AB$ measures 4 cm and is a diameter of circle $P$. In triangle $ABC$, point $C$ is on circle $P$ and $BC = 2$ cm. What is the area of the shaded region?
+128408 Since the base of triangle ACB is a diameter, then ACB is a right angle
And we can use the Pythagorean Theorem to find AC
AC = √[AB^2 - BC^2] = √[4^2 - 2^2 ] = √12 units^2 = 2√3 cm
And the area of triangle ABC = 1/2 the product of the leg lengths =
1/2 * AC * BC = 1/2 (2√3) ( 2) = 2√3 cm^2 (1)
And the area of the circle = pi * (diameter /2)^2 = pi * (4/2)^2 = pi * 2^2 = 4 pi cm^2 (2)
So....the shaded area =
area of the circle - area of triangle ABC =
(2) - (1) =
[ 4pi - 2√3 ) cm^2 ≈ 9.10 cm^2
