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.
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
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"
To determine the measure of the central angle, we can use the formula:
Central Angle = (Length of Chord / Radius) * 360 degrees
Given that the length of the chord intercepted by the central angle is 16 inches and the radius of the circle is 20 inches, we can substitute these values into the formula:
Central Angle = (16 inches / 20 inches) * 360 degrees
Simplifying the expression:
Central Angle = (4/5) * 360 degrees
Central Angle = 4 * 72 degrees
Central Angle = 288 degrees
Therefore, the measure of the central angle is 288 degrees.