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 M with \( m \angle LMN = 60^\circ \) and \( LM = 15 \) units, find the length of arc \( LN \). Round to the nearest hundredth.
**Diagram Description:**
The accompanying diagram depicts a circle named "M" with its center labeled as "M." Two points "L" and "N" lie on the circumference of the circle, forming the arc \( LN \). The angle formed at the center of the circle, denoted as \( \angle LMN \), has a measure of \( 60^\circ \). The line segment joining points L and M is 15 units in length.
**Solution Approach:**
1. To find the length of arc \( LN \), we need to use the formula for the length of an arc:
\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
\]
where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.
2. Here, \( \theta = 60^\circ \) and \( r = 15 \) units.
3. Substituting the values into the formula gives:
\[
\text{Arc Length} = \frac{60^\circ}{360^\circ} \times 2\pi \times 15
\]
Simplifying the fraction \( \frac{60^\circ}{360^\circ} \):
\[
\frac{60}{360} = \frac{1}{6}
\]
4. Therefore:
\[
\text{Arc Length} = \frac{1}{6} \times 2\pi \times 15 = \frac{1}{6} \times 30\pi
\]
5. Simplifying further:
\[
\text{Arc Length} = 5\pi
\]
6. To round to the nearest hundredth, we need to calculate \( 5\pi \) numerically:
\[
5\pi \approx 5 \times 3.14159 \approx 15.70795
\]
7. Rounding to the nearest hundredth:
\[
\text{Arc Length} \approx 15.71 \, \text{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53198e60-c01b-4c66-91c6-7d18bef4fe43%2Fc809a06e-bb07-464a-9243-8f103405e8e8%2F1ds7pt_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
