In circle JJ, mangle, K, J, L, equals, 160, degrees∠KJL=160 ∘ and the area of shaded sector = start fraction, 16, divided by, 9, end fraction, pi 916 π. Find the length of arc, K, L KL⌢ . Express your answer as a fraction times piπ.
Question
In circle JJ, mangle, K, J, L, equals, 160, degrees∠KJL=160 ∘ and the area of shaded sector = start fraction, 16, divided by, 9, end fraction, pi 916 π. Find the length of arc, K, L KL⌢ . Express your answer as a fraction times piπ.
Solution
To solve this problem, we need to follow these steps:
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First, we need to find the radius of the circle. We know that the area of the sector is 16/9 π, and the formula for the area of a sector is (θ/360) * π * r², where θ is the angle of the sector. We know that θ = 160 degrees, so we can set up the equation (160/360) * π * r² = 16/9 π.
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Simplifying the equation, we get r² = (16/9) * (360/160) = 4. Therefore, the radius r = 2.
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Now, we need to find the length of the arc KL. The formula for the length of an arc is (θ/360) * 2πr. Substituting the known values, we get (160/360) * 2π * 2 = 32/9 π.
So, the length of arc KL is 32/9 π.
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