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 )

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

Area_shade = Area_circle/2 - Area_triangle

= (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 90^o 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.