In circle TT, mangle, U, T, V, equals, 120, degrees∠UTV=120 ∘ and the area of shaded sector = 3, pi3π. Find the length of arc, U, W, V UWV⌢ . Express your answer as a fraction times piπ.
Question
In circle TT, mangle, U, T, V, equals, 120, degrees∠UTV=120 ∘ and the area of shaded sector = 3, pi3π. Find the length of arc, U, W, V UWV⌢ . Express your answer as a fraction times piπ.
Solution
The problem states that ∠UTV = 120° in circle TT and the area of the shaded sector is 3π. We are asked to find the length of arc UWV.
Step 1: Determine the ratio of ∠UTV to a full circle. A full circle is 360°, so the ratio is 120°/360° = 1/3.
Step 2: Determine the radius of the circle. The area of a sector is given by (angle/360) * π * r². We know the area is 3π and the angle ratio is 1/3, so we can set up the equation (1/3) * π * r² = 3π. Solving for r, we get r = √9 = 3.
Step 3: Find the length of the arc. The length of an arc is given by (angle/360) * 2πr. Using the radius we found and the angle ratio, we get (1/3) * 2π * 3 = 2π.
So, the length of arc UWV is 2π.
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