In the figure above, J L

In the given triangle, we have:

1. $JL = 7$ (the length of the segment from $J$ to $L$),
2. $KN$ is one-fourth the length of $JN$,
3. $\triangle KLM$ and $\triangle KJN$ are right triangles.

To find the length of $LM$, let's denote the following lengths:

• $JN = x$
• $KN = \frac{1}{4} \times JN = \frac{x}{4}$

Since $\triangle KJN$ and $\triangle KLM$ are similar triangles (due to the right angles and the shared angle $K$), the ratio of corresponding sides will be the same.

Using the similarity of triangles: $\frac{KN}{KJ} = \frac{LM}{JL}$

Substitute the known values: $\frac{\frac{x}{4}}{x} = \frac{LM}{7}$

Simplify the left side: $\frac{1}{4} = \frac{LM}{7}$

Now, solve for $LM$: $LM = \frac{7}{4}$

Thus, the length of segment $LM$ is $\frac{7}{4}$ or 1.75 units.

Would you like any further details or have any questions about this solution?

Here are 5 related questions for further practice:

1. What is the length of $JN$?
2. Find the area of $\triangle KJN$.
3. If the length of $JL$ was doubled, how would that affect $LM$?
4. What is the perimeter of $\triangle KLM$?
5. How would you calculate the length of segment $KM$?

Tip: When working with similar triangles, always set up a ratio between corresponding sides to find unknown lengths.