A circle is inscribed in a parallelogram. One angle of the parallelogram measures 60. What are the measures of the four arcs between consecutive points of tangency? Explain

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

Step-by-step 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.

b

step-by-step explanation: