Answer:
The dimensions are;
m1∠60° = π·r/3
m2∠120° = 2/3·π·r
m3∠60° = π·r/3
m4∠120° = 2/3·π·r
Explanation:
Here we have that the parallelogram inscribed in a circle that is the circle is circumscribed by the parallelogram
Hence, the sides of the parallelogram form tangents with circle
From circle theorem we have the size of the four arcs are dependent on the angles formed by the radii lines to the point of contact of the tangent
That is Arc length ∝ angle subtended at center
Since the sum of the interior angles of a parallelogram = 360°
And 2 the two opposite angles in a parallelogram are equal where one angle = 60°, we have;
60° + 60° + θ + θ = 360°
∴ 2·θ = 360° - 120° = 240°
θ = 240°/2 = 120°
The sum of the angle circumscribing a circle and the angle it subtends at the center = 180°
Therefore, the angle 60°of the parallelogram subtends angle 180 - 60 or 120° at the center of the circle
Similarly, the angle 120° of the parallelogram subtends angle 180 - 120 or 60° at the center of the circle
Hence the measures of the four arcs are;
m1∠60°, m2∠120°, m3∠60°, m4∠120°
With dimensions
Length of m1∠60° = 2·π·r×60/360 = π·r/3
Where:
r = radius of the circle
similarly m2∠120° = 2/3·π·r
m3∠60° = π·r/3
m4∠120° = 2/3·π·r.