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.
![### Geometry Problem: Area of a Sector
#### Problem Statement
In circle \( K \) with \( m \angle JKL = 112 \) and \( JK = 13 \) units, find the area of sector \( JKL \). Round to the nearest hundredth.
#### Diagram Description
The diagram features a circle with center \( K \). Point \( J \) and point \( L \) lie on the circumference of the circle. The radius \( JK \) measures 13 units. The angle \( \angle JKL \) formed by radii \( JK \) and \( KL \) is 112 degrees. The circular sector \( JKL \) is shaded to indicate the area to be found.
#### Solution Process
To find the area of sector \( JKL \), use the formula for the area of a sector of a circle:
\[ \text{Area} = \pi r^2 \times \frac{\theta}{360} \]
Where:
- \( r \) is the radius of the circle.
- \( \theta \) is the central angle in degrees.
Given:
- \( r = 13 \) units
- \( \theta = 112^\circ \)
Substitute these values into the formula:
\[ \text{Area} = \pi \times 13^2 \times \frac{112}{360} \]
\[ \text{Area} = \pi \times 169 \times \frac{112}{360} \]
\[ \text{Area} = \pi \times 169 \times 0.3111 \]
\[ \text{Area} \approx 530.66 \times 0.3111 \]
\[ \text{Area} \approx 165.10 \]
Hence, the area of sector \( JKL \) is approximately \( 165.10 \) square units.
#### Answer
\[ 165.10 \] square units
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