To find the perimeter of the shaded region, we need to first find the radius of the sector FOG.
Since the angle FOG is 1.2 radians and the circumference of a full circle is 2Ï€r, the circumference of the sector FOG can be calculated as:
circumference of sector FOG = (1.2 / 2Ï€) * 2Ï€ * r = 1.2r
Since the circumference is equal to the arc length of the sector, we have:
1.2r = 5
r = 5 / 1.2 = 25 / 6 ≈ 4.17 cm
The perimeter of the shaded region consists of the arc length of the sector FOG and the segment EH. The arc length of the sector FOG is 1.2 * 4.17 ≈ 5.00 cm.
To find the length of segment EH, we can use the fact that the area of the sector EOH is 71.4 cm^2. The ratio of the area of the sector EOH to the area of the full circle is equal to the ratio of the angle EOH to 2Ï€ radians:
(area of sector EOH) / (Ï€ * 5^2) = angle EOH / 2Ï€
71.4 / 25Ï€ = angle EOH / 2Ï€
angle EOH = 2Ï€ * 71.4 / 25Ï€ = 4.54 radians
The length of segment EH can now be calculated as:
length of EH = 5 * 4.54 / 2Ï€ = 3.64 cm (approximately)
Therefore, the total perimeter of the shaded region is approximately 5.00 cm (arc length of sector FOG) + 3.64 cm (length of segment EH) = 8.64 cm.