Question
In circle L with m, angle, K, L, M, equals, 90, degreesm∠KLM=90
∘
and K, L, equals, 6KL=6 units, find the length of arc, K, M
KM
⌢
. Round to the nearest hundredth.
∘
and K, L, equals, 6KL=6 units, find the length of arc, K, M
KM
⌢
. Round to the nearest hundredth.
Answers
GPT-4o mini
To find the length of the arc \( \overset{\frown}{KM} \) in circle \( L \), we first need to find the measure of the angle that subtends the arc at the center of the circle (which is angle \( KLM \)).
1. **Find the radius of the circle**: The arc \( \overset{\frown}{KM} \) is subtended by angle \( KLM = 90^\circ \), and we know \( KL = 6 \) units. Since \( L \) is the center of the circle, \( KL \) is the radius of the circle \( L \). Therefore:
\[ r = KL = 6 \text{ units} \]
2. **Find the circumference of the circle**: The circumference \( C \) of a circle is given by the formula:
\[
C = 2\pi r
\]
Substituting the radius into the formula:
\[
C = 2\pi(6) = 12\pi \text{ units}
\]
3. **Determine the fraction of the circle represented by the angle**: The angle \( KLM = 90^\circ \) is a quarter of a full circle. Therefore, the fraction of the circle for \( \overset{\frown}{KM} \) is:
\[
\text{Fraction} = \frac{90^\circ}{360^\circ} = \frac{1}{4}
\]
4. **Calculate the length of arc \( \overset{\frown}{KM} \)**: The length of the arc can be calculated by taking the fraction of the circumference that corresponds to the angle \( KLM \):
\[
\text{Length of arc } \overset{\frown}{KM} = \text{Fraction} \times C = \frac{1}{4} \times 12\pi
\]
Simplifying this:
\[
\text{Length of arc } \overset{\frown}{KM} = 3\pi \text{ units}
\]
5. **Approximate the length**: Using \( \pi \approx 3.14 \), we can find an approximate value:
\[
3\pi \approx 3 \times 3.14 \approx 9.42 \text{ units}
\]
Therefore, the length of arc \( \overset{\frown}{KM} \) is approximately \( \boxed{9.42} \) units when rounded to the nearest hundredth.
1. **Find the radius of the circle**: The arc \( \overset{\frown}{KM} \) is subtended by angle \( KLM = 90^\circ \), and we know \( KL = 6 \) units. Since \( L \) is the center of the circle, \( KL \) is the radius of the circle \( L \). Therefore:
\[ r = KL = 6 \text{ units} \]
2. **Find the circumference of the circle**: The circumference \( C \) of a circle is given by the formula:
\[
C = 2\pi r
\]
Substituting the radius into the formula:
\[
C = 2\pi(6) = 12\pi \text{ units}
\]
3. **Determine the fraction of the circle represented by the angle**: The angle \( KLM = 90^\circ \) is a quarter of a full circle. Therefore, the fraction of the circle for \( \overset{\frown}{KM} \) is:
\[
\text{Fraction} = \frac{90^\circ}{360^\circ} = \frac{1}{4}
\]
4. **Calculate the length of arc \( \overset{\frown}{KM} \)**: The length of the arc can be calculated by taking the fraction of the circumference that corresponds to the angle \( KLM \):
\[
\text{Length of arc } \overset{\frown}{KM} = \text{Fraction} \times C = \frac{1}{4} \times 12\pi
\]
Simplifying this:
\[
\text{Length of arc } \overset{\frown}{KM} = 3\pi \text{ units}
\]
5. **Approximate the length**: Using \( \pi \approx 3.14 \), we can find an approximate value:
\[
3\pi \approx 3 \times 3.14 \approx 9.42 \text{ units}
\]
Therefore, the length of arc \( \overset{\frown}{KM} \) is approximately \( \boxed{9.42} \) units when rounded to the nearest hundredth.