https://file.aiquickdraw.com/mathbot/file/1730289075848egpwnsou.jpg

To solve question 4, let's analyze the information provided for triangle \( \triangle LMN \):

- Given values:
  - \( LM = 15.5 \, \text{cm} \)
  - \( \angle LNM = 90^\circ \) (right triangle)
  - \( HM = 9.2 \, \text{cm} \)
  - \( \angle HMN = 38^\circ \)

We need to find:
1. \( \angle MLN \)
2. The length of \( HL \)

### Solution

#### Part (i): Finding \( \angle MLN \)

In \( \triangle HMN \), we know that:
\[
\angle LNM = 90^\circ
\]
Using the triangle sum property:
\[
\angle MLN + \angle HMN + \angle LNM = 180^\circ
\]
Substitute the known angles:
\[
\angle MLN + 38^\circ + 90^\circ = 180^\circ
\]
Solving for \( \angle MLN \):
\[
\angle MLN = 180^\circ - 128^\circ = 52^\circ
\]

So, \( \angle MLN = 52^\circ \).

#### Part (ii): Finding the length of \( HL \)

We can use trigonometry in \( \triangle LMN \) to find \( HL \).

Since \( \angle HMN = 38^\circ \) and \( HM = 9.2 \, \text{cm} \):
1. \( HM \) is the opposite side relative to \( \angle HMN \).
2. \( HL \) is the adjacent side relative to \( \angle HMN \).

Using the tangent function:
\[
\tan(38^\circ) = \frac{HM}{HL}
\]
Substitute the known values:
\[
\tan(38^\circ) = \frac{9.2}{HL}
\]
\[
HL = \frac{9.2}{\tan(38^\circ)}
\]

Calculating this:
\[
HL \approx \frac{9.2}{0.7813} \approx 11.77 \, \text{cm}
\]

So, the length of \( HL \approx 11.77 \, \text{cm} \).

---

Would you like a further breakdown of each step, or do you have any other questions?

### Related Questions

1. How would you solve for \( NL \) in \( \triangle LMN \)?
2. What other trigonometric functions can be used to verify \( HL \)?
3. Can you find the length of \( MN \) in \( \triangle LMN \)?
4. How can you find the area of \( \triangle LMN \)?
5. What would change in the solution if \( \angle HMN \) were different?

### Tip
In right triangles, the tangent function is particularly useful for relating the opposite and adjacent sides relative to an angle.

Learn how to solve a right triangle LMN with given side lengths and angles. Find the unknown angle MLN and side length HL using trigonometric functions and the triangle sum property. Suitable for students in Grades 9-10.

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