Question

1. (a)

(b) the perimeter of a sector of another circle is 14.8 cm.
the angle of the sector is 1.7 radians.
calculate the area of the sector.
(i) ∠coe = 68°×2 = 156° (…)
∠edc = 156°÷2 = 78° (…)
(ii) ∠coe (reflex) = 360° - 156° = 224°
∠cde = 224°÷2 = 112° (…)
(iii) ∠cte = 360° - 90° - 90° - … = 44° (…)
(iv) ∠ocd = 90 - 32° = 58° (…)

Explanation:

Step1: Recall the formula for the perimeter of a sector

The perimeter \( P \) of a sector with radius \( r \) and angle \( \theta \) (in radians) is given by \( P = 2r + r\theta \), where \( 2r \) is the sum of the two radii and \( r\theta \) is the length of the arc.
Given \( P = 14.8 \) cm and \( \theta = 1.7 \) radians, we substitute into the formula:
\( 14.8 = 2r + 1.7r \)

Step2: Solve for the radius \( r \)

Combine like terms:
\( 14.8 = (2 + 1.7)r \)
\( 14.8 = 3.7r \)
Divide both sides by 3.7:
\( r=\frac{14.8}{3.7}=4 \) cm

Step3: Recall the formula for the area of a sector

The area \( A \) of a sector with radius \( r \) and angle \( \theta \) (in radians) is given by \( A=\frac{1}{2}r^{2}\theta \)

Step4: Calculate the area

Substitute \( r = 4 \) cm and \( \theta = 1.7 \) radians into the formula:
\( A=\frac{1}{2}\times(4)^{2}\times1.7 \)
First, calculate \( (4)^{2}=16 \)
Then, \( \frac{1}{2}\times16 = 8 \)
Finally, \( 8\times1.7 = 13.6 \) \( cm^{2} \)

Answer:

The area of the sector is \( 13.6 \) square centimeters.