FOG is a sector of a circle, centre O, with angle FOG = 1.2 radians. EOH a sector of a circle, centre O, with radius 5 cm.

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 2πr, the circumference of the sector FOG can be calculated as:

circumference of sector FOG = (1.2 / 2π) * 2π * 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 2π radians:

(area of sector EOH) / (π * 5^2) = angle EOH / 2π

71.4 / 25π = angle EOH / 2π

angle EOH = 2π * 71.4 / 25π = 4.54 radians

The length of segment EH can now be calculated as:

length of EH = 5 * 4.54 / 2π = 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.