Draw the altitude from circle center to middle of chord
then
sin of half central angle = 8/20
half of central angle = 23.6
central angle = 47.2 deg
Determine the measure of the central angle if the length of the chord intercepted by the central angle of a circle of radius 20 inches is 16 inches.
2 answers
The relation between the chord length a and the central angle α is given by:
a = 2 r sin θ / 2
In this case a = 16 in , r = 20 in
16 = 2 ∙ 20 sin θ / 2
16 = 40 sin θ / 2
Divide both sides by 40
16 / 40 = sin θ / 2
8 ∙ 2 / 8 ∙ 5 = sin θ / 2
2 / 5 = sin θ / 2
sin θ / 2 = 2 / 5
θ / 2 = sin⁻¹ ( 2 / 5 )
θ / 2 = arcsin ( 2 / 5 )
θ / 2 = 23.5781784782°
θ = 2 ∙ 23.5781784782°
θ = 47.1563569564°
θ = 47° 9' 23"
a = 2 r sin θ / 2
In this case a = 16 in , r = 20 in
16 = 2 ∙ 20 sin θ / 2
16 = 40 sin θ / 2
Divide both sides by 40
16 / 40 = sin θ / 2
8 ∙ 2 / 8 ∙ 5 = sin θ / 2
2 / 5 = sin θ / 2
sin θ / 2 = 2 / 5
θ / 2 = sin⁻¹ ( 2 / 5 )
θ / 2 = arcsin ( 2 / 5 )
θ / 2 = 23.5781784782°
θ = 2 ∙ 23.5781784782°
θ = 47.1563569564°
θ = 47° 9' 23"