Step 1: Find the radius of the circle. The perimeter (circumference) of the circle is given as $88$ cm. The formula for the circumference of a circle is $C = 2\pi r$. We are given $C = 88$ cm and $\pi = \frac{22}{7}$. $$ 88 = 2 \times \frac{22}{7} \times r $$ $$ 88 = \frac{44}{7} r $$ To find $r$, multiply both sides by $\frac{7}{44}$: $$ r = 88 \times \frac{7}{44} $$ $$ r = 2 \times 7 $$ $$ r = 14 \text{ cm} $$ Step 2: Calculate the length of the chord AB. The chord AB subtends an angle of $70^\circ$ at the centre of the circle. Let the centre be O. In $\triangle OAB$, $OA = OB = r = 14$ cm. We can use the formula for the length of a chord: $AB = 2r \sin\left(\frac{\theta}{2}\right)$. Here, $r = 14$ cm and $\theta = 70^\circ$. $$ AB = 2 \times 14 \times \sin\left(\frac{70^\circ}{2}\right) $$ $$ AB = 28 \sin(35^\circ) $$ Using a calculator for $\sin(35^\circ)$: $$ \sin(35^\circ) \approx 0.573576436 $$ $$ AB = 28 \times 0.573576436 $$ $$ AB \approx 16.06013999 $$ Rounding to three decimal places: $$ AB \approx 16.060 \text{ cm} $$ The length of the chord AB is $\boxed{16.060 \text{ cm}}$. Step 3: Calculate the area of the minor sector OAB. The formula for the area of a sector is $A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$. Here, $\theta = 70^\circ$, $r = 14$ cm, and $\pi = \frac{22}{7}$. $$ A_{\text{sector}} = \frac{70}{360} \times \frac{22}{7} \times (14)^2 $$ $$ A_{\text{sector}} = \frac{7}{36} \times \frac{22}{7} \times 196 $$ $$ A_{\text{sector}} = \frac{22 \times 196}{36} $$ $$ A_{\text{sector}} = \frac{22 \times 49}{9} $$ $$ A_{\text{sector}} = \frac{1078}{9} $$ $$ A_{\text{sector}} \approx 119.777777... \text{ cm}^2 $$ Step 4: Calculate the area of the triangle OAB. The formula for the area of a triangle with two sides and the included angle is $A_{\triangle} = \frac{1}{2}ab\sin(C)$. In $\triangle OAB$, $OA = OB = r = 14$ cm, and the included angle $\angle AOB = 70^\circ$. $$ A_{\triangle OAB} = \frac{1}{2} r^2 \sin(\theta) $$ $$ A_{\triangle OAB} = \frac{1}{2} (14)^2 \sin(70^\circ) $$ $$ A_{\triangle OAB} = \frac{1}{2} \times 196 \times \sin(70^\circ) $$ $$ A_{\triangle OAB} = 98 \sin(70^\circ) $$ Using a calculator for $\sin(70^\circ)$: $$ \sin(70^\circ) \approx 0.939692621 $$ $$ A_{\triangle OAB} = 98 \times 0.939692621 $$ $$ A_{\triangle OAB} \approx 92.109876858 \text{ cm}^2 $$ Step 5: Calculate the area of the minor segment. The area of the minor segment is the area of the sector OAB minus the area of the triangle OAB. $$ A_{\text{segment}} = A_{\text{sector}} - A_{\triangle OAB} $$ $$ A_{\text{segment}} = \frac{1078}{9} - 98 \sin(70^\circ) $$ $$ A_{\text{segment}} \approx 119.777777... - 92.109876858 $$ $$ A_{\text{segment}} \approx 27.667900142 \text{ cm}^2 $$ Rounding to three decimal places: $$ A_{\text{segment}} \approx 27.668 \text{ cm}^2 $$ The area of the minor segment is $\boxed{27.668 \text{ cm}^2}$.