Identify the knowns and the unknown in the right-angled triangle.

We are given:

• Angle $B = 30^\circ$.
• Side $BC = 12$ (this is the side adjacent to angle B). We need to find:
• Side $AC$ (this is the side opposite to angle B).

Since we have the adjacent side and need the opposite side relative to angle B, we use the tangent function:

We know that $\tan 30^\circ = \frac{1}{\sqrt{3}}$.

Multiply both sides by 12:

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{3}$:

The exact length of AC is $\boxed{4\sqrt{3}}$.

4. Q is due East of R and due South of P. The bearing of R from Q is $270^\circ$. Show this information on the diagram.

Since no diagram is provided for this question, I will describe how to represent this information.

Conversely, R is directly to the West of Q.

Conversely, P is directly to the North of Q.

From Q, a bearing of $270^\circ$ is directly West.

To show this on a diagram:

• Draw point Q.
• Draw a North line from Q.
• Draw a line segment extending West from Q, and label the endpoint R. This represents R being due West of Q (or Q being due East of R).
• Draw a line segment extending North from Q, and label the endpoint P. This represents P being due North of Q (or Q being due South of P).
• Indicate the angle of $270^\circ$ clockwise from the North line at Q to the line segment QR.