Let's solve this step by step.

Step 1: Analyze the given information

• We are given a triangle $KLM$, with the measure of $\angle K = 30^\circ$.
• $KL = 10$ units.
• $\tan(\angle M) = \frac{1}{2}$.

We need to find the length of $KM$.

Step 2: Draw a diagram

Draw a triangle $KLM$. Place $K$ at the origin of a coordinate system, with $L$ and $M$ at two other points. Now, draw the altitude from point $L$ perpendicular to $KM$, meeting $KM$ at point $P$. This divides $\triangle KLM$ into two right triangles, $\triangle KLP$ and $\triangle LPM$.

Step 3: Use trigonometric relations

In $\triangle KLP$:

• Since $\angle K = 30^\circ$, and we know $KL = 10$, we can find the length of the altitude $LP$ and the projection $KP$ using basic trigonometry.

• For the altitude $LP$, we use: $\sin(30^\circ) = \frac{LP}{KL}$ $\sin(30^\circ) = \frac{LP}{10}$ Since $\sin(30^\circ) = \frac{1}{2}$: $\frac{1}{2} = \frac{LP}{10} \quad \Rightarrow \quad LP = 5 \text{ units}.$

• For the base $KP$, we use: $\cos(30^\circ) = \frac{KP}{KL}$ $\cos(30^\circ) = \frac{KP}{10}$ Since $\cos(30^\circ) = \frac{\sqrt{3}}{2}$: $\frac{\sqrt{3}}{2} = \frac{KP}{10} \quad \Rightarrow \quad KP = 5\sqrt{3} \text{ units}.$

In $\triangle LPM$:

• From the given $\tan(\angle M) = \frac{1}{2}$, we know: $\tan(\angle M) = \frac{LP}{PM} = \frac{5}{PM}.$ Therefore, $PM = 10$ units.

Step 4: Find the length of $KM$

Now, we can find $KM$ by adding the lengths of $KP$ and $PM$: $KM = KP + PM = 5\sqrt{3} + 10.$ Thus, the length of $KM$ is $5\sqrt{3} + 10$ units.

Summary

The length of $KM$ is $5\sqrt{3} + 10$ units.

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Would you like further clarification on any step, or do you have any questions?

Here are five follow-up questions to consider:

1. What is the geometric significance of the altitude in a triangle?
2. How can you use the tangent function to find other unknown sides in a right triangle?
3. How would the problem change if $\tan(M)$ had a different value?
4. What would be the impact if $\angle K$ were greater than $30^\circ$?
5. Can the Pythagorean theorem be used here? If so, how?

Tip: In any triangle, altitudes are useful tools for creating right triangles and applying trigonometric functions effectively.