Answer:
Tthe length of AB is 13.6 cm
Step-by-step explanation:
Step 1: calculate length CD using the area of a triangle
[tex]A = \frac{1}{2} \times (AD \times CD) \times \ sine (D)\\\\52 = \frac{1}{2} \times (12 \times CD) \times \ sine (102)\\\\52 = 6CD \times 0.978\\\\52 = 5.868CD\\\\CD = \frac{52}{5.868} \\\\CD = 8.862 \ cm[/tex]
Step 2: calculate length AC using Cosine rule;
[tex]AC^2 = AD^2 + CD^2 - 2(AD \times CD)cos (D)\\\\AC^2 = 12^2 + 8.862^2 - 2(12 \times 8.862)cos(102)\\\\AC^2 = 222.535 - 2(-22.11)\\\\AC^2 = 222.535 + 44.22\\\\AC^2 = 266.755\\\\AC = \sqrt{266.755} \\\\AC = 16.33 \ cm[/tex]
Step 3: Apply sine rule to calculate length AB;
[tex]\frac{AB}{sin \ 46} = \frac{AC}{sin \ 120} \\\\AB = \frac{sin \ 46 \ \ \times \ \ AC}{sin \ 120} \\\\AB = \frac{sin \ 46 \ \ \times \ \ 16.33}{sin \ 120} \\\\AB = 13.56 \ cm \\\\AB = 13.6 \ cm[/tex]
Therefore, the length of AB is 13.6 cm
