In the diagram, A, B and C are points in the same horizontal plane - NSC Mathematics - Question 7 - 2023 - Paper 2

Using the cosine rule in triangle ABC:

$$AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(100°)$$

Substituting AC = 10\sqrt{3} and BC = 8:

$$AB^2 = (10\sqrt{3})^2 + 8^2 - 2 \cdot (10\sqrt{3}) \cdot 8 \cdot \cos(100°)$$

Carry out the calculations:

$$AB^2 = 300 + 64 + 16\sqrt{3} \approx 20.3 \text{ units}$$

To find the size of ∠DBA, we can use the sine formula in triangle ABD:

$$\frac{AB}{\sin(∠ABD)} = \frac{AD}{\sin(∠DBA)}$$

Substituting the known values:

$$\frac{20.3}{\sin(73.4°)} = \frac{20}{\sin(∠DBA)}$$

Cross-multiplying gives:

$$sin(∠DBA) = \frac{20 \cdot \sin(73.4°)}{20.3}$$

Calculating gives:

$$∠DBA \approx 76.58°$$