Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![**Problem 11: Finding the Area of a Shaded Circle Segment**
**Objective:**
Determine the area of the shaded segment of a circle. Round the answer to the nearest tenth.
**Diagram Description:**
- A circle is illustrated with center \( O \).
- Two radii \( OM \) and \( OL \) intersect the circle, creating a sector \( MOL \).
- The angle between the two radii \( OM \) and \( OL \) is labeled as \( 120^\circ \).
- The radius of the circle is given as \( 9 \) cm.
- The area of the sector \( MOL \) is partially shaded.
**Steps to Find the Area of the Shaded Segment:**
1. **Calculate the Area of the Sector:**
The area of the sector with angle \( \theta \) (in degrees) is given by the formula:
\[
\text{Area of sector} = \pi r^2 \left(\frac{\theta}{360}\right)
\]
In this problem:
\[
r = 9 \text{ cm}, \quad \theta = 120^\circ
\]
Thus, the area of the sector \( MOL \):
\[
\text{Area of sector} = \pi (9)^2 \left(\frac{120}{360}\right)
\]
Simplify:
\[
\text{Area of sector} = \pi (81) \left(\frac{1}{3}\right) = 27\pi \text{ cm}^2
\]
2. **Calculate the Area of the Triangular Portion \( \Delta MOL \):**
The area of the triangle can be found using trigonometric relationships or by using the formula for the area of a triangle when two sides and the included angle are known. Given the radius is \( 9 \) cm and the included angle is \( 120^\circ \):
\[
\text{Area of } \Delta MOL = \frac{1}{2} r^2 \sin(\theta)
\]
Substitute \( r = 9 \text{ cm}, \quad \theta = 120^\circ \):
\[
\text{Area of } \Delta MOL = \frac{1}{2} (9)^2 \sin(120^\circ)
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