(a) To find the perimeter of the shaded segment, we first determine the radius of the circle. The perimeter of the sector is given by the sum of the arc length and twice the radius:
\(2r + 8 = 20\)
Solving for \(r\), we get \(r = 6\) cm.
The angle \(\angle AOB\) in radians is given by:
\(\frac{8}{6} = \frac{4}{3} \text{ radians}\)
Using the cosine rule in triangle \(OAB\), the length of \(AB\) is:
\(AB = \sqrt{6^2 + 6^2 - 2 \times 6 \times 6 \times \cos \left( \frac{4}{3} \right)}\)
Calculating this gives \(AB \approx 7.42\) cm.
Thus, the perimeter of the shaded segment is:
\(7.42 + 8 = 15.4 \text{ cm}\)
(b) To find the area of the shaded segment, we calculate the area of the sector and subtract the area of triangle \(OAB\).
The area of the sector is:
\(\frac{1}{2} \times 6^2 \times \frac{4}{3}\)
The area of triangle \(OAB\) is:
\(\frac{1}{2} \times 6 \times 6 \times \sin \left( \frac{4}{3} \right)\)
Calculating these gives the sector area as \(24\) cm2 and the triangle area as \(17.49\) cm2.
Thus, the area of the shaded segment is:
\(24 - 17.49 = 6.51 \text{ cm}^2\)
