Question

what is the value of k?
l
8
m
4
k
n
m

Explanation:

Step1: Apply geometric mean theorem

In a right - triangle, if an altitude is drawn to the hypotenuse, then the length of the altitude is the geometric mean between the lengths of the segments of the hypotenuse. Also, each leg of the right - triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Another property: If in a right - triangle \( \triangle LNM\) with right - angle at \(N\) and altitude \(NO\) to hypotenuse \(LM\), then \(NO^{2}=LO\times MO\). But here, we can use the property of similar triangles.
Let the right - triangle \( \triangle LNM\) with right - angle at \(N\) and \(NO\perp LM\). We know that \( \triangle LNO\sim\triangle NMO\sim\triangle LNM\).
For similar triangles \( \triangle LNO\) and \( \triangle NMO\), we have the proportion \(\frac{NO}{LO}=\frac{MO}{NO}\). Also, a more straightforward formula: If in a right - triangle, the length of the altitude \(h\) to the hypotenuse \(c\) divides the hypotenuse into two segments \(x\) and \(y\) (\(c=x + y\)), and we have the relation that if one of the sub - triangles (formed by the altitude) has a side related to the segments of the hypotenuse. In our case, using the property \(NO^{2}=LO\times MO\) (where \(NO = 4\), \(LO = 8\), and \(MO=k\)).

Step2: Solve for \(k\)

Substitute the values into the formula \(4^{2}=8\times k\).
We get \(16 = 8k\).
Divide both sides of the equation by \(8\): \(k=\frac{16}{8}\).

So, \(k = 2\).

Answer:

\(k = 2\)