Here is a shaded shape ABCD - Edexcel - GCSE Maths - Question 17 - 2018 - Paper 3

In triangle OAD, we can use the sine rule to find the length of side OD:

Using the angles and side AD:

$\frac{AD}{\sin(\text{Angle OAD})} = \frac{OD}{\sin(\text{Angle AOD})}$

Substituting the known values:

$\frac{14}{\sin(24°)} = \frac{OD}{\sin(140°)}$

Calculating OD:

$OD = 14 \times \frac{\sin(140°)}{\sin(24°)} \approx 34.25 \text{ cm}.$

In triangle OAD, we can also find length OA using the cosine rule:

$AC^2 = AD^2 + OD^2 - 2 \cdot AD \cdot OD \cdot \cos(\text{Angle AOD})$

Calculating AC:

$AC \approx (14^2 + (34.25)^2 - 2 \cdot 14 \cdot 34.25 \cdot \cos(140°))$

which calculates to approximately 27.32 cm.

Finally, we can sum the lengths of the sides and the arc to find the total perimeter:

$\text{Perimeter} = AD + CD + OD + AC = 14 + 20.66 + 34.25 + 27.32 \approx 96.23 \text{ cm}.$

Rounded to three significant figures, the perimeter is approximately:

$96.2 ext{ cm}.$