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Let's go through the solution to find the lengths of \( MK \) and \( JK \) step by step based on the given diagram and dataLet's go through the solution to find the lengths of \( MK \) and \( JK \) step by step based on the given diagram and 

### Information from the diagram:
- \( LJ = 13 \, \text{m} \)
- \( LM = 5 \, \text{m} \)
- \( KN = 10 \, \text{m} \)
- \( \angle JML = 90^\circ \)
- \( \angle KNM = 30^\circ \)
- \( MJ \) is vertical, so \( \angle MJL = 90^\circ \), implying \( JML \) is a right-angled triangle.
- \( LMN \) is a straight line.

### (i) Calculate the length of \( MK \)

The triangle \( KMN \) is a right-angled triangle with \( \angle KNM = 30^\circ \) and \( KN = 10 \, \text{m} \). We can use trigonometry to calculate \( MK \).

From the sine rule in a right-angled triangle:
\[
\sin(30^\circ) = \frac{MK}{KN}
\]
Since \( \sin(30^\circ) = 0.5 \) and \( KN = 10 \, \text{m} \), we can find \( MK \) as:
\[
MK = 10 \times 0.5 = 5 \, \text{m}
\]

Thus, \( MK = 5 \, \text{m} \).

### (ii) Calculate the length of \( JK \)

Now, consider the triangle \( JMK \), where \( \angle JML = 90^\circ \) and \( LJ = 13 \, \text{m} \).

To find \( JK \), we use the Pythagorean theorem in triangle \( JML \) as:
\[
JL^2 = JM^2 + ML^2
\]
Given \( JL = 13 \, \text{m} \) and \( ML = 5 \, \text{m} \):
\[
13^2 = JM^2 + 5^2
\]
\[
169 = JM^2 + 25
\]
\[
JM^2 = 169 - 25 = 144
\]
\[
JM = 12 \, \text{m}
\]

Now, using triangle \( JMK \), where \( JM = 12 \, \text{m} \) and \( MK = 5 \, \text{m} \), we apply the Pythagorean theorem again:
\[
JK^2 = JM^2 + MK^2
\]
\[
JK^2 = 12^2 + 5^2 = 144 + 25 = 169
\]
\[
JK = \sqrt{169} = 13 \, \text{m}
\]

Thus, \( JK = 13 \, \text{m} \).

### Final Answers:
(i) \( MK = 5 \, \text{m} \)  
(ii) \( JK = 13 \, \text{m} \)

Would you like further clarification on any steps or details?

### 5 Follow-up Questions:
1. How is the sine function used to solve for \( MK \) in the right-angled triangle \( KMN \)?
2. Why is the Pythagorean theorem applicable in the triangles \( JML \) and \( JMK \)?
3. What other trigonometric functions could we have used to find \( MK \)?
4. How does knowing \( \angle KNM = 30^\circ \) simplify the problem for \( MK \)?
5. Could we solve for \( JK \) using trigonometry instead of the Pythagorean theorem?

### Tip:
Always check if the problem can be simplified by recognizing special right triangles, such as \( 30^\circ \)-\( 60^\circ \)-\( 90^\circ \), which often appear in geometry problems.

Calculate the lengths of MK and JK based on the given right-angled triangle diagram, where LJ = 13 m, LM = 5 m, KN = 10 m, and angle KNM = 30°.

Learn how to calculate the lengths of sides MK and JK using trigonometric ratios and the Pythagorean Theorem in a right-angled triangle. This problem involves calculating lengths with given measurements: LJ = 13 m, LM = 5 m, KN = 10 m, and angle KNM = 30°. Suitable for Grade 9 students.

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