Question

applying the geometric mean (altitude) theorem
what is the value of ( k )?
( k = )

Explanation:

Step1: Apply the geometric mean (altitude) theorem

The geometric mean (altitude) theorem states that in a right - triangle, the altitude \(h\) to the hypotenuse is the geometric mean between the segments of the hypotenuse. Also, for a triangle with an altitude \(NO = 4\) (altitude to the side \(ML\) where \(ML=k + 8\)), and sides \(NM = 7\), \(NL=m\), \(MO=k\), \(OL = 8\). The formula from the geometric mean (altitude) theorem for the segments of the side (if we consider the relationship between the altitude and the segments of the side) is \(NO^{2}=MO\times OL\).

Step2: Substitute the values

We know that \(NO = 4\), \(OL = 8\), and \(MO=k\). Substituting into the formula \(NO^{2}=MO\times OL\), we get \(4^{2}=k\times8\).

Step3: Solve for \(k\)

$$
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$$

Answer:

\(k = 2\)