Determine if the 2 triangles are congruent.
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Then
which is a disk of radius 1, hence with area .
The area of the circular sector is
From that, the equation for the central angle is
The perimeter of the circular sector is the sum of the arc length and twice the radius.
From that, the equation for the central angle is
The value of r can be found by equating these two expressions for θ.
By any of several means, you can find that r=0.2 or 1.25. The latter value corresponds to the requirement that the central angle be less than 90°. The radius is ...
1.25 miles.
From that, the equation for the central angle is
The perimeter of the circular sector is the sum of the arc length and twice the radius.
From that, the equation for the central angle is
The value of r can be found by equating these two expressions for θ.
By any of several means, you can find that r=0.2 or 1.25. The latter value corresponds to the requirement that the central angle be less than 90°. The radius is ...
1.25 miles.