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131 ft.
You would form a triangle with sides of 200 ft (radius of the lawn), 200 ft (radius of the lawn), and x ft (the distance around the sidewalk from one path to the other). Since two of the paths form a 75° angle, the length of the third side (x) can be calculated using the law of cosines:
x^2 = 200^2 + 200^2 - 2(200)(200)cos(75°)
x^2 = 40,000 + 40,000 - 80,000(cos75°)
x^2 = 80,000 - 80,000(0.258819)
x^2 = 80,000 - 20703.52
x^2 = 59296.48
x ≈ 243.5 ft
And since you asked for the distance rounded to the nearest foot, the answer would be 244 ft.