In this problem, we are given a triangle $\triangle KLM$, with $LN$ as the altitude. The altitude means that $LN$ is perpendicular to the base $KM$, dividing $KM$ into two segments: $KN$ and $NM$. Our task is to find the value of $x$ and the lengths of $KN$ and $KM$.

Information provided:

• $KN = 4x - 2$
• $NM = 3x$
• $LN = 5x + 15$

Since $LN$ is the altitude, and it's perpendicular to $KM$, we can apply the Pythagorean theorem to two right triangles formed: $\triangle LKN$ and $\triangle LNM$.

Step 1: Apply Pythagoras in $\triangle LKN$

In this triangle, the legs are $LN$ and $KN$, and the hypotenuse is $LK$.

$LK^2 = LN^2 + KN^2$

Substitute the values: $LK^2 = (5x + 15)^2 + (4x - 2)^2$

Step 2: Apply Pythagoras in $\triangle LNM$

Similarly, in $\triangle LNM$, the legs are $LN$ and $NM$, and the hypotenuse is $LM$.

$LM^2 = LN^2 + NM^2$

Substitute the values: $LM^2 = (5x + 15)^2 + (3x)^2$

Since the two triangles share the same altitude, the total length of $KM = KN + NM$.

Now let's solve these equations to find $x$. I'll compute this next.It appears that the system of equations does not provide a solution for $x$ based on the method I initially attempted. Let me try a different approach. Since $KM$ is the sum of $KN$ and $NM$, I will assume $KN + NM = KM$ holds true based on the total line segment. I will focus on solving $KN + NM$ using geometric properties.

Let's try another method to resolve $x$.It seems the attempt to solve using geometric assumptions was flawed. Instead, let's clarify the steps:

• The equation $KN + NM = KM$ simplifies our work. Since the lengths of $KN$ and $NM$ are given in terms of $x$, we can calculate the total length of $KM$, which is simply:

$KM = KN + NM = (4x - 2) + (3x) = 7x - 2$

This represents the length of the base $KM$. No need for additional Pythagorean equations unless more dimensions are specified.

Would you like me to compute specific measures for $KN$ and $KM$ once we solve for $x$, or clarify further on the geometrical properties?