The area of the required triangle is <u>9/π²(3 + √3)</u> sq. units.
In the question, we are given that a triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 3, 4, and 5.
We are asked for the area of the triangle.
Now, the circumference of the circle = 3 + 4 + 5 = 12 units.
The formula for the circumference is 2πr, which gives is:
2πr = 12,
or, r = 6/π.
The length of the arcs are proportional to its central angle, making the angles: 3θ, 4θ, and 5θ, which needs to sum up to 360°, giving us θ = 360°/12 = 30°.
Thus, the three arcs subtends angles of θ₁ = 3θ = 90°,θ₂ = 4θ = 120°, and θ₃ = 5θ = 150°.
The area of the circle can be calculated as:
Area = (1/2)r² sin θ₁ + (1/2)r² sin θ₂ + (1/2)r² sin θ₃ = r²/2(sin θ₁ + sin θ₂ + sin θ₃).
Substituting the values, we get:
Area = 36/2π²(sin 90° + sin 120° + sin 150°),
or, Area = 36/2π²( 1 + √3/2 + 1/2),
or, Area = 9/π²(3 + √3).
Thus, the area of the required triangle is <u>9/π²(3 + √3)</u> sq. units.
Learn more about a triangle at
brainly.com/question/13734546
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