Answer:
a. 8.94 cm b. 127° c. 12.37 cm²
Step-by-step explanation:
a. Since CM = 4 cm is the perpendicular bisector of AB = 16 cm and r is the radius off the circle. From Pythagoras' theorem,
r² = (AB/2)² + CM²
r = √[(AB/2)² + CM²]
Substituting the values of the variables into r, we have
r = √[(16/2)² + 4²]
r = √[8² + 4²]
r = 4√[2² + 1²]
r = 4√[4 + 1]
r = 4√5 cm
r = 8.94 cm
b. We know that the length of a chord L = 2rsin(θ/2) where r is the radius of the circle, and θ is the angle subtended by the chord AB.
Since L = 2rsin(θ/2) and L = AB = 16 cm,
L/2r = sin(θ/2)
taking sine inverse of both sides, we have
θ/2 = sin⁻¹(L/2r)
multiplying both sides by 2, we have
θ = 2sin⁻¹(L/2r)
substituting the values of the variables, we have
θ = 2sin⁻¹[16/(2 × 8.94)]
θ = 2sin⁻¹[16/17.88]
θ = 2sin⁻¹[0.8949]
θ = 2 × 63.49°
θ = 126.98°
θ ≅ 127°
c. The area of a segment A is given by
A = (θπ/360 - sinθ)r²/2 where θ is the angle subtended by the segment and r = radius of the circle
since θ ≅ 127° and r = 4√5 cm, substituting these values into A, we have
A = (127°π/360 - sin127°)(4√5)²/2
A = (398.93/360 - 0.7988)40
A = (1.1081 - 0.7988)40
A = 0.3093 × 40
A = 12.372 cm²
A ≅ 12.37 cm²
