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 \( KLM \)).
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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:
\[ r = KL = 6 \text{ units} \]
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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:
\[ C = 2\pi(6) = 12\pi \text{ units} \]
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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:
\[ \text{Fraction} = \frac{90^\circ}{360^\circ} = \frac{1}{4} \]
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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:
\[ \text{Length of arc } \overset{\frown}{KM} = 3\pi \text{ units} \]
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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.