https://file.aiquickdraw.com/mathbot/file/1725983185613hom4nyok.jpg

Let's solve the problem step-by-step.

### Part (i): Find the size of ∠BOC

To find the size of ∠BOC, we need to determine the angle between lines OB and OC. The directions of these lines are given:

- Line OB is at 090°.
- Line OC is at 193°.

To find the angle ∠BOC, we subtract the smaller angle from the larger angle:

\[
\angle BOC = 193^\circ - 90^\circ = 103^\circ
\]

So, the size of ∠BOC is **103°**.

### Part (ii): Find the distance between points B and C using the cosine rule

The cosine rule for a triangle with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively, is given by:

\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]

In this problem:
- \(a = 50\) m (OB)
- \(b = 53\) m (OC)
- \(C = 103^\circ\) (∠BOC)

Let's substitute these values into the formula:

\[
BC^2 = 50^2 + 53^2 - 2 \times 50 \times 53 \times \cos(103^\circ)
\]

Calculating step-by-step:

1. Calculate \(50^2 = 2500\)
2. Calculate \(53^2 = 2809\)
3. Calculate \(2 \times 50 \times 53 = 5300\)
4. Calculate \(\cos(103^\circ)\)

Let's compute the exact distance \(BC\) using these values.The distance between points B and C, calculated using the cosine rule, is approximately **81 meters** (rounded to the nearest meter).

### Summary:
1. **Angle ∠BOC** is **103°**.
2. **Distance between points B and C** is **81 meters**.

Would you like further details on any of the steps? Here are some related questions you might find interesting:

1. How to calculate angles between vectors in 2D geometry?
2. What is the cosine rule, and where is it used in geometry?
3. Can you explain the sine rule and its applications?
4. How to find the area of a triangle given the lengths of all three sides?
5. How to convert between degrees and radians?

**Tip:** Remember that angles can be measured in both degrees and radians. Understanding both units is essential for trigonometry and calculus.

Find the size of ∠BOC and the distance between points B and C using the cosine rule.

Learn how to find the angle ∠BOC and the distance between points B and C using the cosine rule. The problem involves calculating angles and applying the cosine rule to determine the distance. Suitable for students in Grades 10-12.

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