https://file.aiquickdraw.com/mathbot/file/1726096559621u7behm4u.jpg

In this triangle, we are tasked with finding two lengths: KL and MN. The key information provided is:

- \( LM = 6.0 \, \text{cm} \)
- \( KN = 5.0 \, \text{cm} \)
- \( \angle LKN = 90^\circ \)
- \( \angle MLN = 130^\circ \)

### Part (a): Finding the length of KL
This triangle contains a right-angle triangle portion at \( \triangle KLN \) since \( \angle LKN = 90^\circ \). We can apply the **Pythagorean theorem** to this right triangle:

\[
KL = \sqrt{KN^2 + LN^2}
\]

To find \( LN \), we first calculate it using the angle \( \angle MLN = 130^\circ \) in triangle \( LMN \), possibly using the **Law of Cosines** since two sides and the included angle are given.

### Part (b): Finding the length of MN
For this part, we will use the **Law of Cosines** because we know two sides and the included angle \( \angle MLN = 130^\circ \):

\[
MN^2 = LM^2 + LN^2 - 2(LM)(LN) \cos(\angle MLN)
\]

Let me now calculate both \( KL \) and \( MN \).It seems like I can’t do more advanced data analysis right now. Please try again later.

If you need, I can guide you through solving the problem step by step manually. Let me know how you'd like to proceed!

Would you like me to explain the steps further?

Here are some related questions:
1. What is the Law of Cosines, and when is it used?
2. How can the Pythagorean Theorem be applied in right triangles?
3. Why is it important to know the angles in a triangle when solving for side lengths?
4. What is the relationship between the sides and angles in non-right triangles?
5. How can trigonometric identities assist in solving geometric problems?

**Tip:** When solving triangles with angles other than 90°, use trigonometric laws like the Law of Cosines or Sines for accurate side-length calculations.

In the figure above, not drawn to scale, LM = 6.0 cm, KN = 5.0 cm, the angle LKN = 90°, and the angle MLN = 130°. Calculate, correct to 1 decimal place: (a) The length of KL, (b) The length of MN.

Learn how to solve triangle KLN with given sides and angles. This problem involves using the Pythagorean Theorem and the Law of Cosines to find the lengths of KL and MN. Suitable for students in Grades 9-12.

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