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.