In circle S with MZRST = 42 and RS Round to the nearest hundredth. = 3 units, find the length of arc RT. %3D R
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 Statement
In circle \( S \) with \( m\angle RST = 42^\circ \) and \( RS = 3 \) units, find the length of arc \( RT \). Round to the nearest hundredth.
### Diagram
Below the problem statement, there is a diagram of a circle labeled \( S \) with points \( R \) and \( T \) on the circumference. Two line segments, \( RS \) and \( ST \), represent radii of the circle and are both 3 units long. The angle between these two radii, \( \angle RST \), is marked as \( 42^\circ \).
#### Circle Diagram Explanation
- Point \( S \) is the center of the circle.
- Point \( R \) is on the circumference of the circle.
- Point \( T \) is also on the circumference of the circle.
- Both \( RS \) and \( ST \) are radii of the circle with a length of 3 units.
- The angle \( \angle RST \) formed by the radii \( RS \) and \( ST \) is \( 42^\circ \).
### Solving the Problem
To find the length of arc \( RT \), you can use the following formula:
\[ \text{Length of Arc} = \left( \frac{\theta}{360} \right) \times 2\pi r \]
where:
- \( \theta \) is the central angle in degrees (in this case, \( 42^\circ \)).
- \( r \) is the radius of the circle (in this case, 3 units).
1. Substitute the values into the formula:
\[
\text{Length of Arc} = \left( \frac{42}{360} \right) \times 2\pi \times 3
\]
2. Simplify inside the parentheses:
\[
\text{Length of Arc} = \left( \frac{42}{360} \right) \times 6\pi
\]
3. Calculate the fraction:
\[
\frac{42}{360} = \frac{7}{60}
\]
4. Substitute back into the equation:
\[
\text{Length of Arc} = \left( \frac{7}{60} \right) \times 6\pi](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe768e832-4394-4037-9c96-0d66378e533e%2F96d07bc8-18fe-4c0b-a300-721c8f6acc9d%2F31buyus_processed.jpeg&w=3840&q=75)
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