Area of Circle and Triangle Example

Question:
In the figure below, ABC is a triangle inscribed in the circle with center O that is on the line AB. If AC = 12, BC = 16, what is the shadded area of the figure above ? ( R is the radius of the circle and PI = 3.14 )
Circle area example
Solution:
Point O is on the line AB and point O is the center of the circle, then AB is a diameter, then angle ACB = 90 degree.
AB2 = AB2 + BC2
(2R)2 = 122 + 162 = 144 + 256 = 400 = 202
so 2R = 20
R = 10
Areashade = Areacircle/2 - Areatriangle
= (1/2)pi R2 - (1/2) 12 × 16
= (1/2)[ 3.14 × 102 - 12 × 16]
= (1/2)[ 3.14 × 100 - 12 × 16]
= (1/2)[314 - 192]
= 61
Therefore, the area of shaded is 61.
Rules used in this example:
1. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two legs.
2. The area of a triangle is equal to one-half the product of the base and the altitude (the base is perpendicular to the altitude).
3. An inscribed angle is an angle whose vertex lies on a circle and whose sides are chords. The inscribed angle is 90o when whose opposite chord is a diameter.
4. The area of a circle is equal to the product of the constant (pi) and the square of the length of the radius of the circle.