https://file.aiquickdraw.com/mathbot/file/1726392964910jvew7uue.jpg

I see three questions related to bearings and distances in the image. Let's work through each of them.

### Question 13:
**Given:**
- \( A \) is 7 km north-east of \( B \).
- \( B \) is 12.5 km north-west of \( C \).

We need to find:
1. The bearing of \( A \) from \( C \).
2. The distance between \( A \) and \( C \) (to the nearest meter).

This problem involves vector geometry, specifically using bearings and distances to form a right triangle.

### Solution for Part 1 (Bearing of \( A \) from \( C \)):
- First, the bearing of \( B \) from \( C \) is given as north-west, which is \( 315^\circ \) (since bearings are measured clockwise from north).
- Next, \( A \) is at a \( 45^\circ \) angle north-east of \( B \), as shown in the diagram.

To find the bearing of \( A \) from \( C \), we add \( 45^\circ \) to the bearing of \( B \) from \( C \) (which is \( 315^\circ \)):
\[
\text{Bearing of } A \text{ from } C = 315^\circ + 45^\circ = 360^\circ.
\]
Therefore, the bearing is \( 360^\circ \), which means directly north.

### Solution for Part 2 (Distance between \( A \) and \( C \)):
Using the Pythagorean theorem, since the triangle formed by \( A \), \( B \), and \( C \) is a right triangle:
\[
AC^2 = AB^2 + BC^2.
\]
Substitute \( AB = 7 \, \text{km} \) and \( BC = 12.5 \, \text{km} \):
\[
AC^2 = 7^2 + 12.5^2 = 49 + 156.25 = 205.25.
\]
Thus:
\[
AC = \sqrt{205.25} \approx 14.33 \, \text{km} = 14,330 \, \text{m}.
\]

So, the distance between \( A \) and \( C \) is approximately **14,330 meters**.

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### Question 14:
**Given:**
- A ship is 5 nautical miles from a wharf on a bearing of \( 321^\circ \).
- A lighthouse is 11.5 nautical

A, B, and C are three towns. A lies 7 km north-east of B, and B lies 12.5 km north-west of C. Find the bearing of A from C. Also, how far is A from C? (Answer to the nearest meter.)

This problem involves calculating the bearing and distance between three towns, A, B, and C. Using bearings, angle addition, and the Pythagorean theorem, we find the bearing of A from C and calculate the distance between A and C. Suitable for high school students in grades 10-12.

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