Question

Help me , it is 50 point ​

Answer 1

(a) 52.0 cm

(b) 62.8 cm

(c) 115 cm

Explanation:

Part (a)

`\textsf{Chord length}=2r \sin \left((\theta)/(2)\right)`

where:

• r = radius
• 
  `\theta` = central angle (measured in degrees)

Given:

• r = 30 cm
• 
  `\theta` = 120°
• chord = PR

Substitute the given values into the formula:

`\begin{aligned}\implies PR & =2(30)\sin \left((120^(\circ))/(2)\right)\\\\& = 30√(3)\\\\ & = 52.0\:\sf cm\:(3\:sf)\end{aligned}`

Part (b)

`\textsf{Arc length}=2 \pi r\left((\theta)/(360^(\circ))\right) \quad`

where:

• r = radius
• 
  `\theta` = central angle (measured in degrees)

Given:

• r = 30 cm
• 
  `\theta` = 120°
• arc = PQR
• 
  `\pi` = 3.142

Substitute the given values into the formula:

`\begin{aligned}\implies PQR & =2 (3.142)(30)\left((120^(\circ))/(360^(\circ))\right) \quad\\\\ & = (1571)/(25)\\\\ & = 62.8 \sf \:\:cm \:(3 \:sf)\end{aligned}`

Part (c)

`\begin{aligned}\textsf{Perimeter of shaded portion} & = \textsf{chord length} + \textsf{arc length}\\\\ & = 30√(3)+(1571)/(25)\\\\ & = 114.80152...\\\\ & = 115\:\: \sf cm\:(3\:sf)\end{aligned}`

Answer 2

a) 52.0 cm

b) 62.8 cm

c) 114.8 cm

Conversion:

120° = 2π/3 radians

(a) Find Length of chord PR;

⇒ 2(radius)sin(θ/2)

⇒ 2(30)sin((2π/3)/2)

⇒ 2(30)sin((2(3.142)/3)/2)

⇒ 30√3

⇒ 52.0 cm

(b) Find Length of arc PQR;

⇒ radius(θ)

⇒ 30(2π/3)

⇒ 30(2(3.142)/3)

⇒ 62.8 cm

(c) The perimeter of shaded region;

⇒ Length of chord + Length of arc

⇒ 62.84 + 51.965

⇒ 114.8 cm