Solve the geometry problems in the diagram provided. Calculate AB, angle ABC, and the area of triangle ABC.

Here are the solutions to the geometry problems.

: Top diagram

a) Calculate the length of AB

Step 1: Calculate the lengths of $AD$ and $CD$ using $\triangle ADC$.
In $\triangle ADC$, $\angle ADC = 90^\circ$, $\angle ACD = 67^\circ$, and $AC = 6$.
We use trigonometric ratios:

$$AD = AC \sin(\angle ACD)$$ $$AD = 6 \sin(67^\circ)$$ $$AD \approx 6 \times 0.9205$$ $$AD \approx 5.523 \text{ units}$$ $$CD = AC \cos(\angle ACD)$$