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.