Question

ABC and ADC are triangles. The area of triangle ADC is 52m^2. Work out the length of AB

Answer 1

Tthe length of AB is 13.6 cm

Explanation:

Step 1: calculate length CD using the area of a triangle

`A = (1)/(2) * (AD * CD) * \ sine (D)\\\\52 = (1)/(2) * (12 * CD) * \ sine (102)\\\\52 = 6CD * 0.978\\\\52 = 5.868CD\\\\CD = (52)/(5.868) \\\\CD = 8.862 \ cm`

Step 2: calculate length AC using Cosine rule;

`AC^2 = AD^2 + CD^2 - 2(AD * CD)cos (D)\\\\AC^2 = 12^2 + 8.862^2 - 2(12 * 8.862)cos(102)\\\\AC^2 = 222.535 - 2(-22.11)\\\\AC^2 = 222.535 + 44.22\\\\AC^2 = 266.755\\\\AC = √(266.755) \\\\AC = 16.33 \ cm`

Step 3: Apply sine rule to calculate length AB;

`(AB)/(sin \ 46) = (AC)/(sin \ 120) \\\\AB = (sin \ 46 \ \ * \ \ AC)/(sin \ 120) \\\\AB = (sin \ 46 \ \ * \ \ 16.33)/(sin \ 120) \\\\AB = 13.56 \ cm \\\\AB = 13.6 \ cm`

Therefore, the length of AB is 13.6 cm